4063:
2412:
2277:
1557:
196:
2012:
1744:
864:
941:
300:
1888:
1630:
1387:
1312:
2086:
2185:
628:
419:
3498:
3952:
3049:
3515:
3291:
2821:
2794:
2769:
estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the HardyâLittlewood
Maximal Function is trivially bounded from
377:
2877:
realized that
Marcinkiewicz's result could greatly simplify his work, at which time he published his former student's theorem together with a generalization of his own.
2578:
1091:
1031:
728:
530:
2847:
2767:
2740:
2713:
2686:
2632:
2605:
2545:
2518:
2491:
1215:
1149:
1122:
1058:
998:
761:
684:
560:
462:
336:
232:
432:(Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on
490:
3615:
2407:{\displaystyle {\frac {1}{p}}={\frac {1-\theta }{p_{0}}}+{\frac {\theta }{p_{1}}},\quad {\frac {1}{q}}={\frac {1-\theta }{q_{0}}}+{\frac {\theta }{q_{1}}}.}
3778:
1468:
4092:
3905:
3760:
3315:
106:
3736:
1906:
1641:
2869:
shortly before he died in World War II. The theorem was almost forgotten by
Zygmund, and was absent from his original works on the theory of
2654:
771:
3098:
1190:
Where
Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the
3510:
3628:
3373:
3717:
3608:
3037:
2994:
2966:
3429:
3987:
93:
883:
3632:
3409:
3389:
3505:
3226:
3783:
3439:
3343:
2436:
3839:
253:
4066:
3788:
3773:
3601:
1774:
3803:
3493:
3363:
1568:
1318:
1243:
4048:
3808:
3457:
2849:
follows immediately from the weak (1,1) estimate and interpolation. The weak (1,1) estimate can be obtained from the
4002:
3926:
3399:
3256:
2870:
4043:
3462:
4087:
3859:
3394:
2456:
2027:
638:
3793:
3467:
3414:
2420:
2129:
572:
57:
3895:
3696:
3488:
3768:
3404:
3992:
3091:
970:
4023:
3967:
3931:
3419:
3348:
3241:
3221:
634:
39:
3246:
3138:
2850:
382:
2463:
4006:
3424:
3310:
3195:
1184:
3972:
3910:
3624:
3580:
3368:
3251:
3190:
3159:
2897:
77:
4097:
3997:
3864:
3550:
3452:
3358:
3305:
3269:
3231:
3164:
3084:
2881:
2799:
2772:
2658:
343:
3977:
3338:
3058:
3033:
2990:
2962:
2940:
2444:
2432:
1894:
1760:
is only assumed to be a quasilinear operator in the following sense: there exists a constant
2550:
1063:
1003:
696:
495:
3982:
3900:
3869:
3849:
3834:
3829:
3824:
3661:
3540:
3479:
3300:
3200:
3155:
3143:
3047:
Zygmund, A. (1956), "On a theorem of
Marcinkiewicz concerning interpolation of operations",
3005:
2930:
2826:
61:
3070:
2745:
2718:
2691:
2664:
2610:
2583:
2523:
2496:
2469:
1193:
1127:
1100:
1036:
976:
739:
662:
538:
435:
314:
210:
3844:
3798:
3746:
3741:
3712:
3593:
3066:
3671:
467:
4033:
3885:
3686:
3353:
3169:
3026:
2980:
2866:
2935:
2918:
4081:
4038:
3962:
3691:
3676:
3666:
3565:
3560:
3545:
3535:
3236:
3150:
3125:
2976:
2452:
1393:
1175:, regular boundedness still holds. To make this more formal, one has to explain that
81:
17:
4028:
3681:
3651:
3322:
3121:
2984:
3957:
3947:
3854:
3656:
3021:
2885:
31:
1552:{\displaystyle \|Tf\|_{r}\leq \gamma N_{p}^{\delta }N_{q}^{1-\delta }\|f\|_{r}}
3890:
3730:
3726:
3722:
3575:
3266:
2639:
3062:
2944:
191:{\displaystyle \lambda _{f}(t)=\omega \left\{x\in X\mid |f(x)|>t\right\}.}
3570:
3555:
1180:
1167:
In other words, even if one only requires weak boundedness on the extremes
2007:{\displaystyle \gamma =2C\left({\frac {r(q-p)}{(r-p)(q-r)}}\right)^{1/r}.}
3210:
3179:
3130:
3107:
2419:
The latter formulation follows from the former through an application of
1739:{\displaystyle \gamma =2\left({\frac {r(q-p)}{(r-p)(q-r)}}\right)^{1/r}.}
657:
47:
859:{\displaystyle \|f\|_{p,w}=\left\||f|^{p}\right\|_{1,w}^{\frac {1}{p}}.}
2190:
A more general formulation of the interpolation theorem is as follows:
46:), is a result bounding the norms of non-linear operators acting on
1900:. The theorem holds precisely as stated, except with Îł replaced by
2888:
published a new proof of the
Marcinkiewicz interpolation theorem.
3597:
3080:
2646:< â. In fact, the Hilbert transform is really unbounded for
2580:. Hence Marcinkiewicz's theorem shows that it is bounded from
3076:
2021:(possibly quasilinear) satisfying an estimate of the form
936:{\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}
641:). The converse is not true. For example, the function 1/
2466:
easily shows that the
Hilbert transform is bounded from
27:
2986:
Elliptic partial differential equations of second order
2520:. A much less obvious fact is that it is bounded from
1756:
A version of the theorem also holds more generally if
3272:
2829:
2802:
2775:
2748:
2721:
2694:
2667:
2613:
2586:
2553:
2526:
2499:
2472:
2280:
2132:
2030:
1909:
1777:
1644:
1571:
1471:
1321:
1246:
1196:
1130:
1103:
1066:
1039:
1006:
979:
886:
774:
742:
699:
665:
575:
541:
498:
470:
438:
385:
346:
317:
256:
213:
109:
3028:
Introduction to
Fourier analysis on Euclidean spaces
4016:
3940:
3919:
3878:
3817:
3759:
3705:
3640:
3528:
3476:
3438:
3382:
3331:
3265:
3209:
3178:
3114:
295:{\displaystyle \lambda _{f}(t)\leq {\frac {C}{t}}.}
3953:Spectral theory of ordinary differential equations
3285:
3025:
2841:
2815:
2788:
2761:
2734:
2707:
2680:
2642:arguments show that it is also bounded for 2 <
2626:
2599:
2572:
2539:
2512:
2485:
2406:
2179:
2080:
2006:
1883:{\displaystyle |T(f+g)(x)|\leq C(|Tf(x)|+|Tg(x)|)}
1882:
1738:
1624:
1551:
1381:
1306:
1209:
1143:
1116:
1085:
1052:
1025:
992:
935:
858:
755:
722:
678:
622:
554:
524:
484:
456:
413:
371:
330:
294:
226:
190:
2923:Proceedings of the American Mathematical Society
1625:{\displaystyle \delta ={\frac {p(q-r)}{r(q-p)}}}
1382:{\displaystyle \|Tf\|_{q,w}\leq N_{q}\|f\|_{q},}
1307:{\displaystyle \|Tf\|_{p,w}\leq N_{p}\|f\|_{p},}
3609:
3092:
2862:
43:
8:
3050:Journal de Mathématiques Pures et Appliquées
2165:
2158:
2143:
2133:
2069:
2062:
2041:
2031:
1749:The constants ÎŽ and Îł can also be given for
1540:
1533:
1482:
1472:
1367:
1360:
1332:
1322:
1292:
1285:
1257:
1247:
782:
775:
608:
601:
583:
576:
393:
386:
354:
347:
64:, but also applies to non-linear operators.
2715:bounds can be derived immediately from the
2081:{\displaystyle \|Tf\|_{q,w}\leq C\|f\|_{p}}
1234:
243:satisfies the following inequality for all
3644:
3616:
3602:
3594:
3516:Vitale's random BrunnâMinkowski inequality
3473:
3099:
3085:
3077:
2180:{\displaystyle \|Tf\|_{q}\leq C\|f\|_{p}.}
623:{\displaystyle \|f\|_{1,w}\leq \|f\|_{1}.}
421:Similarly the space is usually denoted by
80:with real or complex values, defined on a
56:Marcinkiewicz' theorem is similar to the
3277:
3271:
2934:
2919:"The Marcinkiewicz interpolation theorem"
2828:
2807:
2801:
2780:
2774:
2753:
2747:
2726:
2720:
2699:
2693:
2672:
2666:
2618:
2612:
2591:
2585:
2558:
2552:
2531:
2525:
2504:
2498:
2477:
2471:
2393:
2384:
2373:
2356:
2343:
2331:
2322:
2311:
2294:
2281:
2279:
2168:
2146:
2131:
2072:
2044:
2029:
1991:
1987:
1927:
1908:
1872:
1852:
1844:
1824:
1810:
1778:
1776:
1723:
1719:
1659:
1643:
1578:
1570:
1543:
1521:
1516:
1506:
1501:
1485:
1470:
1370:
1354:
1335:
1320:
1295:
1279:
1260:
1245:
1201:
1195:
1135:
1129:
1108:
1102:
1071:
1065:
1044:
1038:
1011:
1005:
984:
978:
925:
915:
909:
891:
885:
842:
831:
820:
815:
806:
785:
773:
747:
741:
714:
709:
700:
698:
670:
664:
611:
586:
574:
546:
540:
502:
497:
474:
469:
437:
396:
384:
357:
345:
322:
316:
279:
261:
255:
218:
212:
169:
152:
114:
108:
3906:Group algebra of a locally compact group
2198:is a quasilinear operator of weak type (
1753: = â by passing to the limit.
1221:but this bound increases to infinity as
2917:Hunt, Richard A.; Weiss, Guido (1964).
2909:
2874:
958:Informally, Marcinkiewicz's theorem is
566:and in addition one has the inequality
239:such that the distribution function of
2439:, the Hilbert transform of a function
309:in the inequality above is called the
2240:, then for each Ξ â (0,1),
873:norm is defined as the best constant
7:
3529:Applications & related
2443:can be computed by first taking the
2431:A famous application example is the
3448:Marcinkiewicz interpolation theorem
2861:The theorem was first announced by
36:Marcinkiewicz interpolation theorem
3374:Symmetric decreasing rearrangement
3278:
3006:"Sur l'interpolation d'operations"
2808:
2781:
2103:. An operator is simply of type (
1237:, Theorem VIII.9.2), suppose that
414:{\displaystyle \|f\|_{1,\infty }.}
403:
25:
2936:10.1090/S0002-9939-1964-0169038-4
2655:HardyâLittlewood maximal function
2115:is a bounded transformation from
1183:subset and can be completed. See
4062:
4061:
3988:Topological quantum field theory
1097:is also a bounded operator from
4093:Theorems in functional analysis
2957:DiBenedetto, Emmanuele (2002),
2342:
3032:, Princeton University Press,
2653:Another famous example is the
1977:
1965:
1962:
1950:
1945:
1933:
1877:
1873:
1869:
1863:
1853:
1845:
1841:
1835:
1825:
1821:
1811:
1807:
1801:
1798:
1786:
1779:
1709:
1697:
1694:
1682:
1677:
1665:
1616:
1604:
1596:
1584:
903:
897:
827:
816:
807:
802:
710:
701:
689:as the space of all functions
656:Similarly, one may define the
519:
507:
451:
439:
273:
267:
170:
166:
160:
153:
126:
120:
1:
3784:Uniform boundedness principle
3344:Convergence almost everywhere
2823:, strong boundedness for all
2865:, who showed this result to
1764: > 0 such that
3511:PrĂ©kopaâLeindler inequality
3364:Locally integrable function
3286:{\displaystyle L^{\infty }}
2871:singular integral operators
2816:{\displaystyle L^{\infty }}
2789:{\displaystyle L^{\infty }}
2661:rather than linear. While
2455:, and finally applying the
1417:, and the operator norm of
532:, which has norm 4 not 2.)
372:{\displaystyle \|f\|_{1,w}}
235:if there exists a constant
4114:
3927:Invariant subspace problem
3257:Square-integrable function
3004:Marcinkiewicz, J. (1939),
2451:, then multiplying by the
1033:and at the same time from
340:and is usually denoted by
4057:
3647:
3506:MinkowskiâSteiner formula
2457:inverse Fourier transform
2427:Applications and examples
3896:Spectrum of a C*-algebra
3489:Isoperimetric inequality
2423:and a duality argument.
1440:interpolation inequality
3993:Noncommutative geometry
3494:BrunnâMinkowski theorem
3024:; Weiss, Guido (1971),
2573:{\displaystyle L^{1,w}}
1086:{\displaystyle L^{q,w}}
1026:{\displaystyle L^{p,w}}
971:bounded linear operator
723:{\displaystyle |f|^{p}}
525:{\displaystyle 1/(1-x)}
40:JĂłzef Marcinkiewicz
4049:TomitaâTakesaki theory
4024:Approximation property
3968:Calculus of variations
3349:Convergence in measure
3287:
3010:C. R. Acad. Sci. Paris
2843:
2842:{\displaystyle p>1}
2817:
2790:
2763:
2736:
2709:
2682:
2628:
2601:
2574:
2541:
2514:
2487:
2408:
2181:
2082:
2008:
1884:
1740:
1626:
1553:
1438:. Then the following
1383:
1308:
1211:
1145:
1118:
1087:
1054:
1027:
994:
937:
860:
757:
724:
680:
639:Chebyshev's Inequality
624:
556:
526:
486:
458:
415:
373:
332:
305:The smallest constant
296:
228:
192:
4044:BanachâMazur distance
4007:Generalized functions
3463:RieszâFischer theorem
3288:
3247:Polarization identity
2851:Vitali covering lemma
2844:
2818:
2791:
2764:
2762:{\displaystyle L^{1}}
2737:
2735:{\displaystyle L^{1}}
2710:
2708:{\displaystyle L^{p}}
2683:
2681:{\displaystyle L^{p}}
2629:
2627:{\displaystyle L^{p}}
2602:
2600:{\displaystyle L^{p}}
2575:
2542:
2540:{\displaystyle L^{1}}
2515:
2513:{\displaystyle L^{2}}
2488:
2486:{\displaystyle L^{2}}
2409:
2182:
2083:
2009:
1885:
1741:
1627:
1554:
1384:
1309:
1212:
1210:{\displaystyle L^{r}}
1179:is bounded only on a
1146:
1144:{\displaystyle L^{r}}
1119:
1117:{\displaystyle L^{r}}
1088:
1055:
1053:{\displaystyle L^{q}}
1028:
995:
993:{\displaystyle L^{p}}
938:
861:
758:
756:{\displaystyle L^{p}}
725:
681:
679:{\displaystyle L^{p}}
625:
557:
555:{\displaystyle L^{1}}
527:
487:
459:
457:{\displaystyle (0,1)}
416:
374:
333:
331:{\displaystyle L^{1}}
297:
229:
227:{\displaystyle L^{1}}
193:
94:distribution function
18:Marcinkiewicz theorem
3789:Kakutani fixed-point
3774:Riesz representation
3468:RieszâThorin theorem
3311:Infimum and supremum
3270:
3196:Lebesgue integration
2863:Marcinkiewicz (1939)
2827:
2800:
2773:
2746:
2719:
2692:
2665:
2611:
2584:
2551:
2524:
2497:
2470:
2278:
2212:) and of weak type (
2130:
2028:
1907:
1775:
1642:
1569:
1469:
1319:
1244:
1225:converges to either
1194:
1185:Riesz-Thorin theorem
1128:
1101:
1064:
1037:
1004:
977:
884:
772:
740:
697:
663:
633:This is nothing but
573:
562:function belongs to
539:
496:
468:
436:
383:
344:
315:
254:
211:
107:
58:RieszâThorin theorem
3973:Functional calculus
3932:Mahler's conjecture
3911:Von Neumann algebra
3625:Functional analysis
3430:Young's convolution
3369:Measurable function
3252:Pythagorean theorem
3242:Parseval's identity
3191:Integrable function
2989:, Springer-Verlag,
2898:Interpolation space
2421:Hölder's inequality
1532:
1511:
1187:for these details.
950: > 0.
869:More directly, the
852:
635:Markov's inequality
485:{\displaystyle 1/x}
247: > 0:
78:measurable function
3998:Riemann hypothesis
3697:Topological vector
3551:Probability theory
3453:Plancherel theorem
3359:Integral transform
3306:Chebyshev distance
3283:
3232:Euclidean distance
3165:Minkowski distance
3053:, NeuviĂšme SĂ©rie,
2981:Trudinger, Neil S.
2839:
2813:
2786:
2759:
2732:
2705:
2678:
2659:sublinear operator
2624:
2597:
2570:
2537:
2510:
2483:
2464:Parseval's theorem
2404:
2177:
2078:
2004:
1880:
1736:
1622:
1549:
1512:
1497:
1379:
1304:
1207:
1141:
1114:
1083:
1050:
1023:
990:
933:
877:in the inequality
856:
800:
753:
720:
676:
620:
552:
522:
482:
454:
411:
369:
328:
292:
224:
188:
4075:
4074:
3978:Integral operator
3755:
3754:
3591:
3590:
3524:
3523:
3339:Almost everywhere
3124: &
2650:equal to 1 or â.
2445:Fourier transform
2433:Hilbert transform
2399:
2379:
2351:
2337:
2317:
2289:
2091:is said to be of
1981:
1713:
1620:
1233:. Specifically (
931:
850:
287:
16:(Redirected from
4105:
4088:Fourier analysis
4065:
4064:
3983:Jones polynomial
3901:Operator algebra
3645:
3618:
3611:
3604:
3595:
3541:Fourier analysis
3499:Milman's reverse
3482:
3480:Lebesgue measure
3474:
3458:RiemannâLebesgue
3301:Bounded function
3292:
3290:
3289:
3284:
3282:
3281:
3201:Taxicab geometry
3156:Measurable space
3101:
3094:
3087:
3078:
3073:
3042:
3031:
3017:
2999:
2971:
2949:
2948:
2938:
2914:
2848:
2846:
2845:
2840:
2822:
2820:
2819:
2814:
2812:
2811:
2795:
2793:
2792:
2787:
2785:
2784:
2768:
2766:
2765:
2760:
2758:
2757:
2741:
2739:
2738:
2733:
2731:
2730:
2714:
2712:
2711:
2706:
2704:
2703:
2687:
2685:
2684:
2679:
2677:
2676:
2657:, which is only
2633:
2631:
2630:
2625:
2623:
2622:
2606:
2604:
2603:
2598:
2596:
2595:
2579:
2577:
2576:
2571:
2569:
2568:
2546:
2544:
2543:
2538:
2536:
2535:
2519:
2517:
2516:
2511:
2509:
2508:
2492:
2490:
2489:
2484:
2482:
2481:
2413:
2411:
2410:
2405:
2400:
2398:
2397:
2385:
2380:
2378:
2377:
2368:
2357:
2352:
2344:
2338:
2336:
2335:
2323:
2318:
2316:
2315:
2306:
2295:
2290:
2282:
2186:
2184:
2183:
2178:
2173:
2172:
2151:
2150:
2087:
2085:
2084:
2079:
2077:
2076:
2055:
2054:
2013:
2011:
2010:
2005:
2000:
1999:
1995:
1986:
1982:
1980:
1948:
1928:
1889:
1887:
1886:
1881:
1876:
1856:
1848:
1828:
1814:
1782:
1745:
1743:
1742:
1737:
1732:
1731:
1727:
1718:
1714:
1712:
1680:
1660:
1631:
1629:
1628:
1623:
1621:
1619:
1599:
1579:
1558:
1556:
1555:
1550:
1548:
1547:
1531:
1520:
1510:
1505:
1490:
1489:
1388:
1386:
1385:
1380:
1375:
1374:
1359:
1358:
1346:
1345:
1313:
1311:
1310:
1305:
1300:
1299:
1284:
1283:
1271:
1270:
1235:DiBenedetto 2002
1216:
1214:
1213:
1208:
1206:
1205:
1150:
1148:
1147:
1142:
1140:
1139:
1123:
1121:
1120:
1115:
1113:
1112:
1092:
1090:
1089:
1084:
1082:
1081:
1059:
1057:
1056:
1051:
1049:
1048:
1032:
1030:
1029:
1024:
1022:
1021:
999:
997:
996:
991:
989:
988:
942:
940:
939:
934:
932:
930:
929:
920:
919:
910:
896:
895:
865:
863:
862:
857:
851:
843:
841:
830:
826:
825:
824:
819:
810:
796:
795:
762:
760:
759:
754:
752:
751:
729:
727:
726:
721:
719:
718:
713:
704:
685:
683:
682:
677:
675:
674:
629:
627:
626:
621:
616:
615:
597:
596:
561:
559:
558:
553:
551:
550:
531:
529:
528:
523:
506:
491:
489:
488:
483:
478:
463:
461:
460:
455:
420:
418:
417:
412:
407:
406:
378:
376:
375:
370:
368:
367:
337:
335:
334:
329:
327:
326:
301:
299:
298:
293:
288:
280:
266:
265:
233:
231:
230:
225:
223:
222:
197:
195:
194:
189:
184:
180:
173:
156:
119:
118:
92:, Ï). The
62:linear operators
38:, discovered by
21:
4113:
4112:
4108:
4107:
4106:
4104:
4103:
4102:
4078:
4077:
4076:
4071:
4053:
4017:Advanced topics
4012:
3936:
3915:
3874:
3840:HilbertâSchmidt
3813:
3804:GelfandâNaimark
3751:
3701:
3636:
3622:
3592:
3587:
3520:
3477:
3472:
3434:
3410:HausdorffâYoung
3390:BabenkoâBeckner
3378:
3327:
3273:
3268:
3267:
3261:
3205:
3174:
3170:Sequence spaces
3110:
3105:
3046:
3040:
3020:
3003:
2997:
2975:
2969:
2956:
2953:
2952:
2916:
2915:
2911:
2906:
2894:
2882:Richard A. Hunt
2859:
2825:
2824:
2803:
2798:
2797:
2776:
2771:
2770:
2749:
2744:
2743:
2722:
2717:
2716:
2695:
2690:
2689:
2668:
2663:
2662:
2634:for any 1 <
2614:
2609:
2608:
2587:
2582:
2581:
2554:
2549:
2548:
2527:
2522:
2521:
2500:
2495:
2494:
2473:
2468:
2467:
2429:
2389:
2369:
2358:
2327:
2307:
2296:
2276:
2275:
2239:
2232:
2225:
2218:
2211:
2204:
2164:
2142:
2128:
2127:
2068:
2040:
2026:
2025:
1949:
1929:
1923:
1922:
1905:
1904:
1773:
1772:
1681:
1661:
1655:
1654:
1640:
1639:
1600:
1580:
1567:
1566:
1539:
1481:
1467:
1466:
1437:
1416:
1366:
1350:
1331:
1317:
1316:
1291:
1275:
1256:
1242:
1241:
1197:
1192:
1191:
1131:
1126:
1125:
1104:
1099:
1098:
1067:
1062:
1061:
1040:
1035:
1034:
1007:
1002:
1001:
980:
975:
974:
956:
921:
911:
887:
882:
881:
814:
805:
801:
781:
770:
769:
743:
738:
737:
708:
695:
694:
666:
661:
660:
607:
582:
571:
570:
542:
537:
536:
494:
493:
466:
465:
434:
433:
392:
381:
380:
353:
342:
341:
318:
313:
312:
257:
252:
251:
214:
209:
208:
139:
135:
110:
105:
104:
70:
28:
23:
22:
15:
12:
11:
5:
4111:
4109:
4101:
4100:
4095:
4090:
4080:
4079:
4073:
4072:
4070:
4069:
4058:
4055:
4054:
4052:
4051:
4046:
4041:
4036:
4034:Choquet theory
4031:
4026:
4020:
4018:
4014:
4013:
4011:
4010:
4000:
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3950:
3944:
3942:
3938:
3937:
3935:
3934:
3929:
3923:
3921:
3917:
3916:
3914:
3913:
3908:
3903:
3898:
3893:
3888:
3886:Banach algebra
3882:
3880:
3876:
3875:
3873:
3872:
3867:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3827:
3821:
3819:
3815:
3814:
3812:
3811:
3809:BanachâAlaoglu
3806:
3801:
3796:
3791:
3786:
3781:
3776:
3771:
3765:
3763:
3757:
3756:
3753:
3752:
3750:
3749:
3744:
3739:
3737:Locally convex
3734:
3720:
3715:
3709:
3707:
3703:
3702:
3700:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3648:
3642:
3638:
3637:
3623:
3621:
3620:
3613:
3606:
3598:
3589:
3588:
3586:
3585:
3584:
3583:
3578:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3532:
3530:
3526:
3525:
3522:
3521:
3519:
3518:
3513:
3508:
3503:
3502:
3501:
3491:
3485:
3483:
3471:
3470:
3465:
3460:
3455:
3450:
3444:
3442:
3436:
3435:
3433:
3432:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3386:
3384:
3380:
3379:
3377:
3376:
3371:
3366:
3361:
3356:
3354:Function space
3351:
3346:
3341:
3335:
3333:
3329:
3328:
3326:
3325:
3320:
3319:
3318:
3308:
3303:
3297:
3295:
3280:
3276:
3263:
3262:
3260:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3227:CauchyâSchwarz
3224:
3218:
3216:
3207:
3206:
3204:
3203:
3198:
3193:
3187:
3185:
3176:
3175:
3173:
3172:
3167:
3162:
3153:
3148:
3147:
3146:
3136:
3128:
3126:Hilbert spaces
3118:
3116:
3115:Basic concepts
3112:
3111:
3106:
3104:
3103:
3096:
3089:
3081:
3075:
3074:
3044:
3038:
3018:
3001:
2995:
2977:Gilbarg, David
2973:
2967:
2961:, BirkhÀuser,
2951:
2950:
2929:(6): 996â998.
2908:
2907:
2905:
2902:
2901:
2900:
2893:
2890:
2875:Zygmund (1956)
2867:Antoni Zygmund
2858:
2855:
2838:
2835:
2832:
2810:
2806:
2783:
2779:
2756:
2752:
2729:
2725:
2702:
2698:
2675:
2671:
2621:
2617:
2594:
2590:
2567:
2564:
2561:
2557:
2534:
2530:
2507:
2503:
2480:
2476:
2435:. Viewed as a
2428:
2425:
2417:
2416:
2415:
2414:
2403:
2396:
2392:
2388:
2383:
2376:
2372:
2367:
2364:
2361:
2355:
2350:
2347:
2341:
2334:
2330:
2326:
2321:
2314:
2310:
2305:
2302:
2299:
2293:
2288:
2285:
2270:
2269:
2237:
2230:
2223:
2216:
2209:
2202:
2188:
2187:
2176:
2171:
2167:
2163:
2160:
2157:
2154:
2149:
2145:
2141:
2138:
2135:
2089:
2088:
2075:
2071:
2067:
2064:
2061:
2058:
2053:
2050:
2047:
2043:
2039:
2036:
2033:
2015:
2014:
2003:
1998:
1994:
1990:
1985:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1947:
1944:
1941:
1938:
1935:
1932:
1926:
1921:
1918:
1915:
1912:
1891:
1890:
1879:
1875:
1871:
1868:
1865:
1862:
1859:
1855:
1851:
1847:
1843:
1840:
1837:
1834:
1831:
1827:
1823:
1820:
1817:
1813:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1781:
1747:
1746:
1735:
1730:
1726:
1722:
1717:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1679:
1676:
1673:
1670:
1667:
1664:
1658:
1653:
1650:
1647:
1633:
1632:
1618:
1615:
1612:
1609:
1606:
1603:
1598:
1595:
1592:
1589:
1586:
1583:
1577:
1574:
1560:
1559:
1546:
1542:
1538:
1535:
1530:
1527:
1524:
1519:
1515:
1509:
1504:
1500:
1496:
1493:
1488:
1484:
1480:
1477:
1474:
1442:holds for all
1433:
1412:
1390:
1389:
1378:
1373:
1369:
1365:
1362:
1357:
1353:
1349:
1344:
1341:
1338:
1334:
1330:
1327:
1324:
1314:
1303:
1298:
1294:
1290:
1287:
1282:
1278:
1274:
1269:
1266:
1263:
1259:
1255:
1252:
1249:
1204:
1200:
1165:
1164:
1138:
1134:
1111:
1107:
1080:
1077:
1074:
1070:
1047:
1043:
1020:
1017:
1014:
1010:
987:
983:
955:
952:
944:
943:
928:
924:
918:
914:
908:
905:
902:
899:
894:
890:
867:
866:
855:
849:
846:
840:
837:
834:
829:
823:
818:
813:
809:
804:
799:
794:
791:
788:
784:
780:
777:
750:
746:
717:
712:
707:
703:
673:
669:
631:
630:
619:
614:
610:
606:
603:
600:
595:
592:
589:
585:
581:
578:
549:
545:
521:
518:
515:
512:
509:
505:
501:
481:
477:
473:
453:
450:
447:
444:
441:
410:
405:
402:
399:
395:
391:
388:
366:
363:
360:
356:
352:
349:
325:
321:
303:
302:
291:
286:
283:
278:
275:
272:
269:
264:
260:
221:
217:
199:
198:
187:
183:
179:
176:
172:
168:
165:
162:
159:
155:
151:
148:
145:
142:
138:
134:
131:
128:
125:
122:
117:
113:
100:is defined by
69:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4110:
4099:
4096:
4094:
4091:
4089:
4086:
4085:
4083:
4068:
4060:
4059:
4056:
4050:
4047:
4045:
4042:
4040:
4039:Weak topology
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4021:
4019:
4015:
4008:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3963:Index theorem
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3945:
3943:
3939:
3933:
3930:
3928:
3925:
3924:
3922:
3920:Open problems
3918:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3883:
3881:
3877:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3822:
3820:
3816:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3766:
3764:
3762:
3758:
3748:
3745:
3743:
3740:
3738:
3735:
3732:
3728:
3724:
3721:
3719:
3716:
3714:
3711:
3710:
3708:
3704:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3649:
3646:
3643:
3639:
3634:
3630:
3626:
3619:
3614:
3612:
3607:
3605:
3600:
3599:
3596:
3582:
3579:
3577:
3574:
3573:
3572:
3569:
3567:
3566:Sobolev space
3564:
3562:
3561:Real analysis
3559:
3557:
3554:
3552:
3549:
3547:
3546:Lorentz space
3544:
3542:
3539:
3537:
3536:Bochner space
3534:
3533:
3531:
3527:
3517:
3514:
3512:
3509:
3507:
3504:
3500:
3497:
3496:
3495:
3492:
3490:
3487:
3486:
3484:
3481:
3475:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3445:
3443:
3441:
3437:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3381:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3336:
3334:
3330:
3324:
3321:
3317:
3314:
3313:
3312:
3309:
3307:
3304:
3302:
3299:
3298:
3296:
3294:
3274:
3264:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3237:Hilbert space
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3219:
3217:
3215:
3213:
3208:
3202:
3199:
3197:
3194:
3192:
3189:
3188:
3186:
3184:
3182:
3177:
3171:
3168:
3166:
3163:
3161:
3157:
3154:
3152:
3151:Measure space
3149:
3145:
3142:
3141:
3140:
3137:
3135:
3133:
3129:
3127:
3123:
3120:
3119:
3117:
3113:
3109:
3102:
3097:
3095:
3090:
3088:
3083:
3082:
3079:
3072:
3068:
3064:
3060:
3056:
3052:
3051:
3045:
3041:
3039:0-691-08078-X
3035:
3030:
3029:
3023:
3019:
3015:
3011:
3007:
3002:
2998:
2996:3-540-41160-7
2992:
2988:
2987:
2982:
2978:
2974:
2970:
2968:3-7643-4231-5
2964:
2960:
2959:Real analysis
2955:
2954:
2946:
2942:
2937:
2932:
2928:
2924:
2920:
2913:
2910:
2903:
2899:
2896:
2895:
2891:
2889:
2887:
2883:
2878:
2876:
2872:
2868:
2864:
2856:
2854:
2852:
2836:
2833:
2830:
2804:
2777:
2754:
2750:
2727:
2723:
2700:
2696:
2673:
2669:
2660:
2656:
2651:
2649:
2645:
2641:
2637:
2619:
2615:
2592:
2588:
2565:
2562:
2559:
2555:
2532:
2528:
2505:
2501:
2478:
2474:
2465:
2460:
2458:
2454:
2453:sign function
2450:
2446:
2442:
2438:
2434:
2426:
2424:
2422:
2401:
2394:
2390:
2386:
2381:
2374:
2370:
2365:
2362:
2359:
2353:
2348:
2345:
2339:
2332:
2328:
2324:
2319:
2312:
2308:
2303:
2300:
2297:
2291:
2286:
2283:
2274:
2273:
2272:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2236:
2233: â
2229:
2222:
2215:
2208:
2201:
2197:
2193:
2192:
2191:
2174:
2169:
2161:
2155:
2152:
2147:
2139:
2136:
2126:
2125:
2124:
2122:
2118:
2114:
2110:
2106:
2102:
2100:
2096:
2073:
2065:
2059:
2056:
2051:
2048:
2045:
2037:
2034:
2024:
2023:
2022:
2020:
2001:
1996:
1992:
1988:
1983:
1974:
1971:
1968:
1959:
1956:
1953:
1942:
1939:
1936:
1930:
1924:
1919:
1916:
1913:
1910:
1903:
1902:
1901:
1899:
1896:
1866:
1860:
1857:
1849:
1838:
1832:
1829:
1818:
1815:
1804:
1795:
1792:
1789:
1783:
1771:
1770:
1769:
1767:
1763:
1759:
1754:
1752:
1733:
1728:
1724:
1720:
1715:
1706:
1703:
1700:
1691:
1688:
1685:
1674:
1671:
1668:
1662:
1656:
1651:
1648:
1645:
1638:
1637:
1636:
1613:
1610:
1607:
1601:
1593:
1590:
1587:
1581:
1575:
1572:
1565:
1564:
1563:
1544:
1536:
1528:
1525:
1522:
1517:
1513:
1507:
1502:
1498:
1494:
1491:
1486:
1478:
1475:
1465:
1464:
1463:
1461:
1458: â
1457:
1453:
1449:
1445:
1441:
1436:
1432:
1428:
1424:
1420:
1415:
1411:
1407:
1403:
1399:
1395:
1394:operator norm
1376:
1371:
1363:
1355:
1351:
1347:
1342:
1339:
1336:
1328:
1325:
1315:
1301:
1296:
1288:
1280:
1276:
1272:
1267:
1264:
1261:
1253:
1250:
1240:
1239:
1238:
1236:
1232:
1228:
1224:
1220:
1202:
1198:
1188:
1186:
1182:
1178:
1174:
1170:
1162:
1158:
1154:
1136:
1132:
1109:
1105:
1096:
1078:
1075:
1072:
1068:
1045:
1041:
1018:
1015:
1012:
1008:
985:
981:
972:
968:
964:
961:
960:
959:
953:
951:
949:
926:
922:
916:
912:
906:
900:
892:
888:
880:
879:
878:
876:
872:
853:
847:
844:
838:
835:
832:
821:
811:
797:
792:
789:
786:
778:
768:
767:
766:
764:
748:
744:
733:
715:
705:
692:
688:
687:
671:
667:
654:
652:
648:
644:
640:
636:
617:
612:
604:
598:
593:
590:
587:
579:
569:
568:
567:
565:
547:
543:
533:
516:
513:
510:
503:
499:
479:
475:
471:
448:
445:
442:
430:
428:
424:
408:
400:
397:
389:
364:
361:
358:
350:
339:
323:
319:
308:
289:
284:
281:
276:
270:
262:
258:
250:
249:
248:
246:
242:
238:
234:
219:
215:
204:
185:
181:
177:
174:
163:
157:
149:
146:
143:
140:
136:
132:
129:
123:
115:
111:
103:
102:
101:
99:
95:
91:
87:
83:
82:measure space
79:
75:
68:Preliminaries
67:
65:
63:
59:
54:
52:
50:
45:
41:
37:
33:
19:
4029:Balanced set
4003:Distribution
3941:Applications
3794:KreinâMilman
3779:Closed graph
3447:
3383:Inequalities
3323:Uniform norm
3211:
3180:
3131:
3054:
3048:
3027:
3022:Stein, Elias
3013:
3009:
2985:
2958:
2926:
2922:
2912:
2879:
2860:
2652:
2647:
2643:
2635:
2461:
2448:
2440:
2430:
2418:
2265:
2261:
2257:
2253:
2249:
2245:
2244:is of type (
2241:
2234:
2227:
2220:
2213:
2206:
2199:
2195:
2189:
2120:
2116:
2112:
2108:
2104:
2098:
2094:
2092:
2090:
2018:
2017:An operator
2016:
1897:
1895:almost every
1892:
1765:
1761:
1757:
1755:
1750:
1748:
1634:
1561:
1459:
1455:
1451:
1447:
1443:
1439:
1434:
1430:
1426:
1422:
1418:
1413:
1409:
1405:
1401:
1397:
1392:so that the
1391:
1230:
1226:
1222:
1218:
1189:
1176:
1172:
1168:
1166:
1160:
1156:
1152:
1094:
966:
962:
957:
947:
945:
874:
870:
868:
735:
731:
690:
658:
655:
650:
646:
642:
632:
563:
534:
431:
426:
422:
310:
306:
304:
244:
240:
236:
206:
202:
200:
97:
89:
85:
73:
71:
55:
48:
35:
29:
3958:Heat kernel
3948:Hardy space
3855:Trace class
3769:HahnâBanach
3731:Topological
3581:Von Neumann
3395:Chebyshev's
3057:: 223â248,
3016:: 1272â1273
2886:Guido Weiss
2268:of the form
2093:weak type (
1429:is at most
1408:is at most
954:Formulation
649:but not to
645:belongs to
32:mathematics
4082:Categories
3891:C*-algebra
3706:Properties
3576:C*-algebra
3400:Clarkson's
2904:References
2437:multiplier
1768:satisfies
734:, and the
730:belong to
693:such that
205:is called
4098:Lp spaces
3865:Unbounded
3860:Transpose
3818:Operators
3747:Separable
3742:Reflexive
3727:Algebraic
3713:Barrelled
3571:*-algebra
3556:Quasinorm
3425:Minkowski
3316:Essential
3279:∞
3108:Lp spaces
3063:0021-7824
2945:0002-9939
2873:. Later
2809:∞
2782:∞
2387:θ
2366:θ
2363:−
2325:θ
2304:θ
2301:−
2166:‖
2159:‖
2153:≤
2144:‖
2134:‖
2070:‖
2063:‖
2057:≤
2042:‖
2032:‖
1972:−
1957:−
1940:−
1911:γ
1816:≤
1704:−
1689:−
1672:−
1646:γ
1611:−
1591:−
1573:δ
1541:‖
1534:‖
1529:δ
1526:−
1508:δ
1495:γ
1492:≤
1483:‖
1473:‖
1368:‖
1361:‖
1348:≤
1333:‖
1323:‖
1293:‖
1286:‖
1273:≤
1258:‖
1248:‖
907:≤
889:λ
783:‖
776:‖
609:‖
602:‖
599:≤
584:‖
577:‖
514:−
464:given by
404:∞
394:‖
387:‖
355:‖
348:‖
277:≤
259:λ
150:∣
144:∈
133:ω
112:λ
4067:Category
3879:Algebras
3761:Theorems
3718:Complete
3687:Schwartz
3633:glossary
3420:Markov's
3415:Hölder's
3405:Hanner's
3222:Bessel's
3160:function
3144:Lebesgue
2983:(2001),
2892:See also
2880:In 1964
2742:to weak
2638:< 2.
2226:) where
1454:and all
1446:between
1217:norm of
1155:between
1151:for any
963:Theorem.
946:for all
828:‖
803:‖
3870:Unitary
3850:Nuclear
3835:Compact
3830:Bounded
3825:Adjoint
3799:Minâmax
3692:Sobolev
3677:Nuclear
3667:Hilbert
3662:Fréchet
3627: (
3440:Results
3139:Measure
3071:0080887
2857:History
2640:Duality
2252:), for
1093:. Then
88:,
42: (
3845:Normal
3682:Orlicz
3672:Hölder
3652:Banach
3641:Spaces
3629:topics
3293:spaces
3214:spaces
3183:spaces
3134:spaces
3122:Banach
3069:
3061:
3036:
2993:
2965:
2943:
2462:Hence
1562:where
765:using
60:about
51:spaces
34:, the
3657:Besov
2260:with
2111:) if
1421:from
1400:from
1181:dense
973:from
969:be a
736:weak
686:space
659:weak
637:(aka
311:weak
207:weak
201:Then
76:be a
4005:(or
3723:Dual
3478:For
3332:Maps
3059:ISSN
3034:ISBN
2991:ISBN
2963:ISBN
2941:ISSN
2884:and
2834:>
2256:and
1893:for
1635:and
1450:and
1171:and
1159:and
965:Let
763:norm
535:Any
492:and
338:norm
175:>
72:Let
44:1939
3014:208
2931:doi
2796:to
2688:to
2607:to
2547:to
2493:to
2447:of
2194:If
2119:to
1425:to
1404:to
1396:of
1229:or
1124:to
1060:to
1000:to
425:or
379:or
96:of
30:In
4084::
3631:â
3067:MR
3065:,
3055:35
3012:,
3008:,
2979:;
2939:.
2927:15
2925:.
2921:.
2853:.
2459:.
2264:â€
2219:,
2205:,
2123::
1462::
653:.
429:.
53:.
4009:)
3733:)
3729:/
3725:(
3635:)
3617:e
3610:t
3603:v
3275:L
3212:L
3181:L
3158:/
3132:L
3100:e
3093:t
3086:v
3043:.
3000:.
2972:.
2947:.
2933::
2837:1
2831:p
2805:L
2778:L
2755:1
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2728:1
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2701:p
2697:L
2674:p
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2648:p
2644:p
2636:p
2620:p
2616:L
2593:p
2589:L
2566:w
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2560:1
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2533:1
2529:L
2506:2
2502:L
2479:2
2475:L
2449:f
2441:f
2402:.
2395:1
2391:q
2382:+
2375:0
2371:q
2360:1
2354:=
2349:q
2346:1
2340:,
2333:1
2329:p
2320:+
2313:0
2309:p
2298:1
2292:=
2287:p
2284:1
2266:q
2262:p
2258:q
2254:p
2250:q
2248:,
2246:p
2242:T
2238:1
2235:q
2231:0
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2224:1
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2217:1
2214:p
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2207:q
2203:0
2200:p
2196:T
2175:.
2170:p
2162:f
2156:C
2148:q
2140:f
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2107:,
2105:p
2101:)
2099:q
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2095:p
2074:p
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2035:T
2019:T
2002:.
1997:r
1993:/
1989:1
1984:)
1978:)
1975:r
1969:q
1966:(
1963:)
1960:p
1954:r
1951:(
1946:)
1943:p
1937:q
1934:(
1931:r
1925:(
1920:C
1917:2
1914:=
1898:x
1878:)
1874:|
1870:)
1867:x
1864:(
1861:g
1858:T
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1850:+
1846:|
1842:)
1839:x
1836:(
1833:f
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1826:|
1822:(
1819:C
1812:|
1808:)
1805:x
1802:(
1799:)
1796:g
1793:+
1790:f
1787:(
1784:T
1780:|
1766:T
1762:C
1758:T
1751:q
1734:.
1729:r
1725:/
1721:1
1716:)
1710:)
1707:r
1701:q
1698:(
1695:)
1692:p
1686:r
1683:(
1678:)
1675:p
1669:q
1666:(
1663:r
1657:(
1652:2
1649:=
1617:)
1614:p
1608:q
1605:(
1602:r
1597:)
1594:r
1588:q
1585:(
1582:p
1576:=
1545:r
1537:f
1523:1
1518:q
1514:N
1503:p
1499:N
1487:r
1479:f
1476:T
1460:L
1456:f
1452:q
1448:p
1444:r
1435:q
1431:N
1427:L
1423:L
1419:T
1414:p
1410:N
1406:L
1402:L
1398:T
1377:,
1372:q
1364:f
1356:q
1352:N
1343:w
1340:,
1337:q
1329:f
1326:T
1302:,
1297:p
1289:f
1281:p
1277:N
1268:w
1265:,
1262:p
1254:f
1251:T
1231:q
1227:p
1223:r
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1203:r
1199:L
1177:T
1173:q
1169:p
1163:.
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1157:p
1153:r
1137:r
1133:L
1110:r
1106:L
1095:T
1079:w
1076:,
1073:q
1069:L
1046:q
1042:L
1019:w
1016:,
1013:p
1009:L
986:p
982:L
967:T
948:t
927:p
923:t
917:p
913:C
904:)
901:t
898:(
893:f
875:C
871:L
854:.
848:p
845:1
839:w
836:,
833:1
822:p
817:|
812:f
808:|
798:=
793:w
790:,
787:p
779:f
749:p
745:L
732:L
716:p
711:|
706:f
702:|
691:f
672:p
668:L
651:L
647:L
643:x
618:.
613:1
605:f
594:w
591:,
588:1
580:f
564:L
548:1
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520:)
517:x
511:1
508:(
504:/
500:1
480:x
476:/
472:1
452:)
449:1
446:,
443:0
440:(
427:L
423:L
409:.
401:,
398:1
390:f
365:w
362:,
359:1
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324:1
320:L
307:C
290:.
285:t
282:C
274:)
271:t
268:(
263:f
245:t
241:f
237:C
220:1
216:L
203:f
186:.
182:}
178:t
171:|
167:)
164:x
161:(
158:f
154:|
147:X
141:x
137:{
130:=
127:)
124:t
121:(
116:f
98:f
90:F
86:X
84:(
74:f
49:L
20:)
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