1940:
1649:
559:
1638:
735:
2501:
969:
3083:
1288:
2927:
1452:
261:
2195:
2073:
Here we use a standard trick involving the maximal function to give a quick proof of
Lebesgue differentiation theorem. (But remember that in the proof of the maximal theorem, we used the Vitali covering lemma.) Let
2323:
1935:{\displaystyle \|Mf\|_{p}^{p}\leq p\int _{0}^{\infty }t^{p-1}\left({2C \over t}\int _{|f|>{\frac {t}{2}}}|f|dx\right)dt=2Cp\int _{0}^{\infty }\int _{|f|>{\frac {t}{2}}}t^{p-2}|f|dxdt=C_{p}\|f\|_{p}^{p}}
2785:
There are several common variants of the Hardy-Littlewood maximal operator which replace the averages over centered balls with averages over different families of sets. For instance, one can define the
2407:
1096:
455:
2599:
2663:
892:
1467:
616:
2707:
2543:
863:
839:
2426:
1994:
2021:
2949:
900:
1160:
2796:
1339:
124:
2092:
2206:
2036:
2051:
2362:
2666:
3266:
1019:
2067:
554:{\displaystyle \left|\{Mf>\lambda \}\right|<{\frac {C_{d}}{\lambda }}\Vert f\Vert _{L^{1}(\mathbf {R} ^{d})}.}
3261:
3135:
2552:
2056:
1633:{\displaystyle \|Mf\|_{p}^{p}=\int \int _{0}^{Mf(x)}pt^{p-1}dtdx=p\int _{0}^{\infty }t^{p-1}|\{Mf>t\}|dt}
730:{\displaystyle \Vert Mf\Vert _{L^{p}(\mathbf {R} ^{d})}\leq C_{p,d}\Vert f\Vert _{L^{p}(\mathbf {R} ^{d})}.}
39:
3256:
2616:
2032:
2024:
803:
570:
868:
2672:
2509:
2047:
Some applications of the Hardy–Littlewood
Maximal Inequality include proving the following results:
844:
820:
2496:{\displaystyle \left|\{\Omega g>\varepsilon \}\right|\leq {\frac {2\,M}{\varepsilon }}\|g\|_{1}}
309:
289:
795:
794: = ∞, the inequality is trivial (since the average of a function is no larger than its
345:
59:
17:
2061:
3240:
47:
1966:
3173:
3112:
2028:
341:
333:
271:
3078:{\displaystyle M_{\Delta }f(x)=\sup _{x\in Q_{x}}{\frac {1}{|Q_{x}|}}\int _{Q_{x}}|f(y)|dy}
1999:
3190:
964:{\displaystyle \bigcup _{B\in {\mathcal {F}}}B\subset \bigcup _{B\in {\mathcal {F'}}}5B}
3208:
749:
1283:{\displaystyle |\{Mf>t\}|\leq 5^{d}\sum _{j}|B_{j}|\leq {5^{d} \over t}\int |f|dy.}
3250:
3236:
43:
3178:
3161:
2922:{\displaystyle f^{*}(x)=\sup _{x\in B_{x}}{\frac {1}{|B_{x}|}}\int _{B_{x}}|f(y)|dy}
3096:
2939:
are required to merely contain x, rather than be centered at x. There is also the
2782:. It is unknown whether there is a weak bound that is independent of dimension.
2746:
about spherical maximal functions can be used to show that, for 1 <
1447:{\displaystyle |\{Mf>t\}|\leq {2C \over t}\int _{|f|>{\frac {t}{2}}}|f|dx,}
1293:
This completes the proof of the weak-type estimate. We next deduce from this the
790:
While there are several proofs of this theorem, a common one is given below: For
256:{\displaystyle Mf(x)=\sup _{r>0}{\frac {1}{|B(x,r)|}}\int _{B(x,r)}|f(y)|\,dy}
2743:
329:
28:
2190:{\displaystyle \Omega f(x)=\limsup _{r\to 0}f_{r}(x)-\liminf _{r\to 0}f_{r}(x)}
806:
to prove the weak-type estimate. (See the article for the proof of the lemma.)
1132:. By the lemma, we can find, among such balls, a sequence of disjoint balls
349:
91:
3202:
The best constant for the centered Hardy–Littlewood maximal inequality
752:
used the Calderón-Zygmund method of rotations to prove the following:
565:
With the Hardy–Littlewood maximal inequality in hand, the following
2318:{\displaystyle f_{r}(x)={\frac {1}{|B(x,r)|}}\int _{B(x,r)}f(y)dy.}
802: < ∞, first we shall use the following version of the
395:). Before stating the theorem more precisely, for simplicity, let {
3231:
Singular
Integrals and Differentiability Properties of Functions
3162:"The development of square functions in the work of A. Zygmund"
3103:. Both of these operators satisfy the HL maximal inequality.
2790:
HL maximal operator (using the notation of Stein-Shakarchi)
944:
917:
875:
850:
826:
437: > 0 such that for all λ > 0 and
1329:/2 and 0 otherwise. By the weak-type estimate applied to
90:
returns the maximum of a set of reals, namely the set of
316:
is finite almost everywhere. This is a corollary of the
2402:{\displaystyle \Omega f\leq \Omega g+\Omega h=\Omega g}
2952:
2799:
2675:
2619:
2555:
2512:
2429:
2365:
2209:
2095:
2002:
1969:
1652:
1470:
1342:
1163:
1022:
903:
871:
847:
823:
619:
458:
127:
2742:
are in the above inequalities. However, a result of
2750: < ∞, we can remove the dependence of
2356:) with norm that can be made arbitrary small. Then
841:a family of open balls with bounded diameter. Then
3077:
2921:
2701:
2657:
2593:
2537:
2495:
2401:
2317:
2189:
2015:
1988:
1934:
1632:
1446:
1282:
1090:
963:
886:
857:
833:
729:
553:
255:
3213:Princeton Lectures in Analysis III: Real Analysis
2979:
2823:
2728:It is still unknown what the smallest constants
2557:
2153:
2115:
741:In the strong type estimate the best bounds for
147:
2709:almost everywhere. By the uniqueness of limit,
2605:. It remains to show the limit actually equals
1091:{\displaystyle \int _{B_{x}}|f|dy>t|B_{x}|.}
2039:below for more about optimizing the constant.
3166:Bulletin of the American Mathematical Society
1109:} is a subset of the union of such balls, as
8:
3204:, Annals of Mathematics, 157 (2003), 647–688
2640:
2620:
2520:
2513:
2484:
2477:
2450:
2435:
1918:
1911:
1663:
1653:
1616:
1601:
1481:
1471:
1363:
1348:
1184:
1169:
690:
683:
630:
620:
569:estimate is an immediate consequence of the
514:
507:
479:
464:
1960:. This completes the proof of the theorem.
2344:is continuous and has compact support and
3177:
3064:
3047:
3039:
3034:
3022:
3016:
3007:
3001:
2993:
2982:
2957:
2951:
2908:
2891:
2883:
2878:
2866:
2860:
2851:
2845:
2837:
2826:
2804:
2798:
2685:
2680:
2674:
2643:
2627:
2618:
2576:
2560:
2554:
2523:
2511:
2487:
2467:
2461:
2428:
2364:
2273:
2261:
2238:
2232:
2214:
2208:
2172:
2156:
2134:
2118:
2094:
2007:
2001:
1980:
1968:
1926:
1921:
1905:
1881:
1873:
1861:
1845:
1837:
1829:
1828:
1818:
1813:
1775:
1767:
1755:
1747:
1739:
1738:
1719:
1702:
1692:
1687:
1671:
1666:
1651:
1619:
1596:
1584:
1574:
1569:
1535:
1510:
1505:
1489:
1484:
1469:
1430:
1422:
1410:
1402:
1394:
1393:
1374:
1366:
1343:
1341:
1266:
1258:
1244:
1238:
1230:
1224:
1215:
1209:
1199:
1187:
1164:
1162:
1080:
1074:
1065:
1048:
1040:
1032:
1027:
1021:
942:
941:
934:
916:
915:
908:
902:
874:
873:
870:
849:
848:
846:
825:
824:
822:
713:
708:
698:
693:
671:
653:
648:
638:
633:
618:
537:
532:
522:
517:
496:
490:
457:
246:
241:
224:
203:
191:
168:
162:
150:
126:
3123:
894:consisting of disjoint balls such that
3241:Topics in Real and Functional Analysis
2613:). But this is easy: it is known that
2594:{\displaystyle \lim _{r\to 0}f_{r}(x)}
3129:
3127:
7:
2778: > 0 only depending on
2658:{\displaystyle \|f_{r}-f\|_{1}\to 0}
1002:, by definition, we can find a ball
430: ≥ 1, there is a constant
3136:"Stein's spherical maximal theorem"
571:Marcinkiewicz interpolation theorem
324:Hardy–Littlewood maximal inequality
318:Hardy–Littlewood maximal inequality
3233:. Princeton University Press, 1971
3220:Maximal functions: spherical means
3215:. Princeton University Press, 2005
2958:
2669:) and thus there is a subsequence
2438:
2393:
2384:
2375:
2366:
2096:
2057:Rademacher differentiation theorem
1819:
1693:
1575:
748:are unknown. However subsequently
25:
2420:and so, by the theorem, we have:
757:Theorem (Dimension Independence).
584: ≥ 1, 1 <
33:Hardy–Littlewood maximal operator
18:Hardy-Littlewood maximal function
3222:, Proc. Natl. Acad. Sci. U.S.A.
2549:= 0 almost everywhere; that is,
2052:Lebesgue differentiation theorem
2031:, and the finite version of the
1996:in the proof can be improved to
817:be a separable metric space and
709:
649:
533:
3179:10.1090/s0273-0979-1982-15040-6
1643:By the estimate above we have:
887:{\displaystyle {\mathcal {F}}'}
578:Theorem (Strong Type Estimate).
3065:
3061:
3055:
3048:
3023:
3008:
2972:
2966:
2909:
2905:
2899:
2892:
2867:
2852:
2816:
2810:
2702:{\displaystyle f_{r_{k}}\to f}
2693:
2649:
2588:
2582:
2564:
2538:{\displaystyle \|g\|_{1}\to 0}
2529:
2303:
2297:
2289:
2277:
2262:
2258:
2246:
2239:
2226:
2220:
2184:
2178:
2160:
2146:
2140:
2122:
2108:
2102:
2068:Fractional integration theorem
1882:
1874:
1838:
1830:
1776:
1768:
1748:
1740:
1620:
1597:
1523:
1517:
1431:
1423:
1403:
1395:
1367:
1344:
1267:
1259:
1231:
1216:
1188:
1165:
1081:
1066:
1049:
1041:
858:{\displaystyle {\mathcal {F}}}
834:{\displaystyle {\mathcal {F}}}
719:
704:
659:
644:
543:
528:
242:
238:
232:
225:
219:
207:
192:
188:
176:
169:
140:
134:
1:
2667:approximation of the identity
2064:on nontangential convergence.
424:Theorem (Weak Type Estimate).
275:-dimensional Lebesgue measure
74:and returns another function
763: ≤ ∞ one can pick
610: > 0 such that
375:) then the maximal function
38:is a significant non-linear
2757:on the dimension, that is,
3283:
3195:Bounded Analytic Functions
865:has a countable subfamily
304:, being the supremum over
300:, so the maximal function
1121:}. This is trivial since
312:. It is not obvious that
288:The averages are jointly
2720:almost everywhere then.
1139:such that the union of 5
798:). For 1 <
3197:. Springer-Verlag, 2006
1989:{\displaystyle C=5^{d}}
1963:Note that the constant
3079:
2923:
2703:
2659:
2601:exists for almost all
2595:
2539:
2497:
2403:
2319:
2191:
2017:
1990:
1936:
1634:
1448:
1284:
1092:
984:
965:
888:
859:
835:
783:
739:
731:
563:
555:
308: > 0, is
257:
3207:Rami Shakarchi &
3160:Stein, E. M. (1982).
3099:containing the point
3080:
2924:
2704:
2660:
2596:
2540:
2498:
2412:by continuity. Now, Ω
2404:
2320:
2192:
2033:Vitali covering lemma
2018:
2016:{\displaystyle 3^{d}}
1991:
1937:
1635:
1449:
1285:
1093:
966:
889:
860:
836:
808:
804:Vitali covering lemma
759:For 1 <
754:
732:
575:
556:
421:
258:
58:The operator takes a
2950:
2943:HL maximal operator
2797:
2673:
2617:
2553:
2510:
2427:
2363:
2207:
2093:
2000:
1967:
1650:
1468:
1340:
1161:
1020:
982:with 5 times radius.
901:
869:
845:
821:
617:
603:there is a constant
588: ≤ ∞, and
456:
363:> 1. That is, if
125:
3200:Antonios D. Melas,
1945:where the constant
1931:
1823:
1697:
1676:
1579:
1527:
1494:
3267:Types of functions
3075:
3000:
2919:
2844:
2771:for some constant
2699:
2655:
2591:
2571:
2535:
2493:
2399:
2315:
2187:
2167:
2129:
2037:Discussion section
2013:
1986:
1932:
1917:
1809:
1683:
1662:
1630:
1565:
1501:
1480:
1444:
1280:
1214:
1088:
961:
954:
923:
884:
855:
831:
796:essential supremum
727:
551:
403:} denote the set {
346:sublinear operator
253:
161:
98:for all the balls
60:locally integrable
3262:Harmonic analysis
3226:(1976), 2174–2175
3028:
2978:
2872:
2822:
2556:
2475:
2267:
2152:
2114:
1853:
1763:
1732:
1418:
1387:
1253:
1205:
930:
904:
505:
197:
146:
78:. For any point
48:harmonic analysis
16:(Redirected from
3274:
3229:Elias M. Stein,
3218:Elias M. Stein,
3184:
3183:
3181:
3157:
3151:
3150:
3148:
3146:
3131:
3113:Rising sun lemma
3095:ranges over all
3084:
3082:
3081:
3076:
3068:
3051:
3046:
3045:
3044:
3043:
3029:
3027:
3026:
3021:
3020:
3011:
3002:
2999:
2998:
2997:
2962:
2961:
2932:where the balls
2928:
2926:
2925:
2920:
2912:
2895:
2890:
2889:
2888:
2887:
2873:
2871:
2870:
2865:
2864:
2855:
2846:
2843:
2842:
2841:
2809:
2808:
2708:
2706:
2705:
2700:
2692:
2691:
2690:
2689:
2664:
2662:
2661:
2656:
2648:
2647:
2632:
2631:
2600:
2598:
2597:
2592:
2581:
2580:
2570:
2544:
2542:
2541:
2536:
2528:
2527:
2506:Now, we can let
2502:
2500:
2499:
2494:
2492:
2491:
2476:
2471:
2462:
2457:
2453:
2408:
2406:
2405:
2400:
2324:
2322:
2321:
2316:
2293:
2292:
2268:
2266:
2265:
2242:
2233:
2219:
2218:
2196:
2194:
2193:
2188:
2177:
2176:
2166:
2139:
2138:
2128:
2029:Lebesgue measure
2025:inner regularity
2022:
2020:
2019:
2014:
2012:
2011:
1995:
1993:
1992:
1987:
1985:
1984:
1952:depends only on
1941:
1939:
1938:
1933:
1930:
1925:
1910:
1909:
1885:
1877:
1872:
1871:
1856:
1855:
1854:
1846:
1841:
1833:
1822:
1817:
1790:
1786:
1779:
1771:
1766:
1765:
1764:
1756:
1751:
1743:
1733:
1728:
1720:
1713:
1712:
1696:
1691:
1675:
1670:
1639:
1637:
1636:
1631:
1623:
1600:
1595:
1594:
1578:
1573:
1546:
1545:
1526:
1509:
1493:
1488:
1453:
1451:
1450:
1445:
1434:
1426:
1421:
1420:
1419:
1411:
1406:
1398:
1388:
1383:
1375:
1370:
1347:
1289:
1287:
1286:
1281:
1270:
1262:
1254:
1249:
1248:
1239:
1234:
1229:
1228:
1219:
1213:
1204:
1203:
1191:
1168:
1125:is contained in
1097:
1095:
1094:
1089:
1084:
1079:
1078:
1069:
1052:
1044:
1039:
1038:
1037:
1036:
970:
968:
967:
962:
953:
952:
951:
950:
922:
921:
920:
893:
891:
890:
885:
883:
879:
878:
864:
862:
861:
856:
854:
853:
840:
838:
837:
832:
830:
829:
736:
734:
733:
728:
723:
722:
718:
717:
712:
703:
702:
682:
681:
663:
662:
658:
657:
652:
643:
642:
560:
558:
557:
552:
547:
546:
542:
541:
536:
527:
526:
506:
501:
500:
491:
486:
482:
419:}. Now we have:
334:J. E. Littlewood
328:This theorem of
262:
260:
259:
254:
245:
228:
223:
222:
198:
196:
195:
172:
163:
160:
110:) of any radius
21:
3282:
3281:
3277:
3276:
3275:
3273:
3272:
3271:
3247:
3246:
3243:(lecture notes)
3191:John B. Garnett
3187:
3159:
3158:
3154:
3144:
3142:
3133:
3132:
3125:
3121:
3109:
3093:
3035:
3030:
3012:
3006:
2989:
2953:
2948:
2947:
2937:
2879:
2874:
2856:
2850:
2833:
2800:
2795:
2794:
2776:
2769:
2762:
2755:
2740:
2733:
2726:
2714:
2681:
2676:
2671:
2670:
2639:
2623:
2615:
2614:
2572:
2551:
2550:
2519:
2508:
2507:
2483:
2463:
2434:
2430:
2425:
2424:
2361:
2360:
2269:
2237:
2210:
2205:
2204:
2168:
2130:
2091:
2090:
2062:Fatou's theorem
2045:
2003:
1998:
1997:
1976:
1965:
1964:
1950:
1901:
1857:
1824:
1734:
1721:
1718:
1714:
1698:
1648:
1647:
1580:
1531:
1466:
1465:
1389:
1376:
1338:
1337:
1297:bounds. Define
1240:
1220:
1195:
1159:
1158:
1154:}. It follows:
1144:
1137:
1130:
1070:
1028:
1023:
1018:
1017:
1007:
943:
899:
898:
872:
867:
866:
843:
842:
819:
818:
788:
777:independent of
775:
768:
746:
707:
694:
689:
667:
647:
634:
629:
615:
614:
608:
531:
518:
513:
492:
463:
459:
454:
453:
435:
326:
199:
167:
123:
122:
86:, the function
56:
23:
22:
15:
12:
11:
5:
3280:
3278:
3270:
3269:
3264:
3259:
3249:
3248:
3245:
3244:
3234:
3227:
3216:
3209:Elias M. Stein
3205:
3198:
3186:
3185:
3172:(2): 359–376.
3168:. New Series.
3152:
3134:Tao, Terence.
3122:
3120:
3117:
3116:
3115:
3108:
3105:
3091:
3086:
3085:
3074:
3071:
3067:
3063:
3060:
3057:
3054:
3050:
3042:
3038:
3033:
3025:
3019:
3015:
3010:
3005:
2996:
2992:
2988:
2985:
2981:
2977:
2974:
2971:
2968:
2965:
2960:
2956:
2935:
2930:
2929:
2918:
2915:
2911:
2907:
2904:
2901:
2898:
2894:
2886:
2882:
2877:
2869:
2863:
2859:
2854:
2849:
2840:
2836:
2832:
2829:
2825:
2821:
2818:
2815:
2812:
2807:
2803:
2774:
2767:
2760:
2753:
2738:
2731:
2725:
2722:
2712:
2698:
2695:
2688:
2684:
2679:
2654:
2651:
2646:
2642:
2638:
2635:
2630:
2626:
2622:
2590:
2587:
2584:
2579:
2575:
2569:
2566:
2563:
2559:
2545:and conclude Ω
2534:
2531:
2526:
2522:
2518:
2515:
2504:
2503:
2490:
2486:
2482:
2479:
2474:
2470:
2466:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2437:
2433:
2410:
2409:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2326:
2325:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2291:
2288:
2285:
2282:
2279:
2276:
2272:
2264:
2260:
2257:
2254:
2251:
2248:
2245:
2241:
2236:
2231:
2228:
2225:
2222:
2217:
2213:
2198:
2197:
2186:
2183:
2180:
2175:
2171:
2165:
2162:
2159:
2155:
2154:lim inf
2151:
2148:
2145:
2142:
2137:
2133:
2127:
2124:
2121:
2117:
2116:lim sup
2113:
2110:
2107:
2104:
2101:
2098:
2071:
2070:
2065:
2059:
2054:
2044:
2041:
2010:
2006:
1983:
1979:
1975:
1972:
1948:
1943:
1942:
1929:
1924:
1920:
1916:
1913:
1908:
1904:
1900:
1897:
1894:
1891:
1888:
1884:
1880:
1876:
1870:
1867:
1864:
1860:
1852:
1849:
1844:
1840:
1836:
1832:
1827:
1821:
1816:
1812:
1808:
1805:
1802:
1799:
1796:
1793:
1789:
1785:
1782:
1778:
1774:
1770:
1762:
1759:
1754:
1750:
1746:
1742:
1737:
1731:
1727:
1724:
1717:
1711:
1708:
1705:
1701:
1695:
1690:
1686:
1682:
1679:
1674:
1669:
1665:
1661:
1658:
1655:
1641:
1640:
1629:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1599:
1593:
1590:
1587:
1583:
1577:
1572:
1568:
1564:
1561:
1558:
1555:
1552:
1549:
1544:
1541:
1538:
1534:
1530:
1525:
1522:
1519:
1516:
1513:
1508:
1504:
1500:
1497:
1492:
1487:
1483:
1479:
1476:
1473:
1455:
1454:
1443:
1440:
1437:
1433:
1429:
1425:
1417:
1414:
1409:
1405:
1401:
1397:
1392:
1386:
1382:
1379:
1373:
1369:
1365:
1362:
1359:
1356:
1353:
1350:
1346:
1291:
1290:
1279:
1276:
1273:
1269:
1265:
1261:
1257:
1252:
1247:
1243:
1237:
1233:
1227:
1223:
1218:
1212:
1208:
1202:
1198:
1194:
1190:
1186:
1183:
1180:
1177:
1174:
1171:
1167:
1142:
1135:
1128:
1099:
1098:
1087:
1083:
1077:
1073:
1068:
1064:
1061:
1058:
1055:
1051:
1047:
1043:
1035:
1031:
1026:
1005:
972:
971:
960:
957:
949:
946:
940:
937:
933:
929:
926:
919:
914:
911:
907:
882:
877:
852:
828:
787:
784:
773:
766:
750:Elias M. Stein
744:
738:
737:
726:
721:
716:
711:
706:
701:
697:
692:
688:
685:
680:
677:
674:
670:
666:
661:
656:
651:
646:
641:
637:
632:
628:
625:
622:
606:
562:
561:
550:
545:
540:
535:
530:
525:
521:
516:
512:
509:
504:
499:
495:
489:
485:
481:
478:
475:
472:
469:
466:
462:
433:
359:to itself for
325:
322:
270:| denotes the
264:
263:
252:
249:
244:
240:
237:
234:
231:
227:
221:
218:
215:
212:
209:
206:
202:
194:
190:
187:
184:
181:
178:
175:
171:
166:
159:
156:
153:
149:
145:
142:
139:
136:
133:
130:
92:average values
55:
52:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3279:
3268:
3265:
3263:
3260:
3258:
3257:Real analysis
3255:
3254:
3252:
3242:
3238:
3237:Gerald Teschl
3235:
3232:
3228:
3225:
3221:
3217:
3214:
3210:
3206:
3203:
3199:
3196:
3192:
3189:
3188:
3180:
3175:
3171:
3167:
3163:
3156:
3153:
3141:
3137:
3130:
3128:
3124:
3118:
3114:
3111:
3110:
3106:
3104:
3102:
3098:
3094:
3072:
3069:
3058:
3052:
3040:
3036:
3031:
3017:
3013:
3003:
2994:
2990:
2986:
2983:
2975:
2969:
2963:
2954:
2946:
2945:
2944:
2942:
2938:
2916:
2913:
2902:
2896:
2884:
2880:
2875:
2861:
2857:
2847:
2838:
2834:
2830:
2827:
2819:
2813:
2805:
2801:
2793:
2792:
2791:
2789:
2783:
2781:
2777:
2770:
2764: =
2763:
2756:
2749:
2745:
2741:
2734:
2723:
2721:
2719:
2715:
2696:
2686:
2682:
2677:
2668:
2652:
2644:
2636:
2633:
2628:
2624:
2612:
2608:
2604:
2585:
2577:
2573:
2567:
2561:
2548:
2532:
2524:
2516:
2488:
2480:
2472:
2468:
2464:
2458:
2454:
2447:
2444:
2441:
2431:
2423:
2422:
2421:
2419:
2415:
2396:
2390:
2387:
2381:
2378:
2372:
2369:
2359:
2358:
2357:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2312:
2309:
2306:
2300:
2294:
2286:
2283:
2280:
2274:
2270:
2255:
2252:
2249:
2243:
2234:
2229:
2223:
2215:
2211:
2203:
2202:
2201:
2181:
2173:
2169:
2163:
2157:
2149:
2143:
2135:
2131:
2125:
2119:
2111:
2105:
2099:
2089:
2088:
2087:
2085:
2081:
2077:
2069:
2066:
2063:
2060:
2058:
2055:
2053:
2050:
2049:
2048:
2042:
2040:
2038:
2034:
2030:
2026:
2023:by using the
2008:
2004:
1981:
1977:
1973:
1970:
1961:
1959:
1955:
1951:
1927:
1922:
1914:
1906:
1902:
1898:
1895:
1892:
1889:
1886:
1878:
1868:
1865:
1862:
1858:
1850:
1847:
1842:
1834:
1825:
1814:
1810:
1806:
1803:
1800:
1797:
1794:
1791:
1787:
1783:
1780:
1772:
1760:
1757:
1752:
1744:
1735:
1729:
1725:
1722:
1715:
1709:
1706:
1703:
1699:
1688:
1684:
1680:
1677:
1672:
1667:
1659:
1656:
1646:
1645:
1644:
1627:
1624:
1613:
1610:
1607:
1604:
1591:
1588:
1585:
1581:
1570:
1566:
1562:
1559:
1556:
1553:
1550:
1547:
1542:
1539:
1536:
1532:
1528:
1520:
1514:
1511:
1506:
1502:
1498:
1495:
1490:
1485:
1477:
1474:
1464:
1463:
1462:
1460:
1441:
1438:
1435:
1427:
1415:
1412:
1407:
1399:
1390:
1384:
1380:
1377:
1371:
1360:
1357:
1354:
1351:
1336:
1335:
1334:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1277:
1274:
1271:
1263:
1255:
1250:
1245:
1241:
1235:
1225:
1221:
1210:
1206:
1200:
1196:
1192:
1181:
1178:
1175:
1172:
1157:
1156:
1155:
1153:
1149:
1145:
1138:
1131:
1124:
1120:
1116:
1112:
1108:
1104:
1085:
1075:
1071:
1062:
1059:
1056:
1053:
1045:
1033:
1029:
1024:
1016:
1015:
1014:
1012:
1008:
1001:
997:
993:
989:
983:
981:
977:
958:
955:
947:
938:
935:
931:
927:
924:
912:
909:
905:
897:
896:
895:
880:
816:
812:
807:
805:
801:
797:
793:
785:
782:
780:
776:
769:
762:
758:
753:
751:
747:
724:
714:
699:
695:
686:
678:
675:
672:
668:
664:
654:
639:
635:
626:
623:
613:
612:
611:
609:
601:
599:
595:
592: ∈
591:
587:
583:
579:
574:
572:
568:
548:
538:
523:
519:
510:
502:
497:
493:
487:
483:
476:
473:
470:
467:
460:
452:
451:
450:
448:
444:
441: ∈
440:
436:
429:
425:
420:
418:
414:
410:
406:
402:
398:
394:
390:
386:
383:-bounded and
382:
378:
374:
370:
366:
362:
358:
356:
352:
347:
343:
339:
335:
331:
323:
321:
319:
315:
311:
307:
303:
299:
295:
291:
286:
284:
280:
276:
274:
269:
250:
247:
235:
229:
216:
213:
210:
204:
200:
185:
182:
179:
173:
164:
157:
154:
151:
143:
137:
131:
128:
121:
120:
119:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
53:
51:
49:
45:
44:real analysis
41:
37:
34:
30:
19:
3230:
3223:
3219:
3212:
3201:
3194:
3169:
3165:
3155:
3143:. Retrieved
3139:
3100:
3097:dyadic cubes
3089:
3087:
2940:
2933:
2931:
2787:
2784:
2779:
2772:
2765:
2758:
2751:
2747:
2736:
2729:
2727:
2717:
2710:
2610:
2606:
2602:
2546:
2505:
2417:
2413:
2411:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2327:
2199:
2083:
2079:
2075:
2072:
2046:
2043:Applications
1962:
1957:
1953:
1946:
1944:
1642:
1458:
1456:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1292:
1151:
1147:
1140:
1133:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1100:
1010:
1009:centered at
1003:
999:
995:
991:
987:
985:
979:
975:
973:
814:
810:
809:
799:
791:
789:
778:
771:
764:
760:
756:
755:
742:
740:
604:
602:
597:
593:
589:
585:
581:
577:
576:
566:
564:
449:), we have:
446:
442:
438:
431:
427:
423:
422:
416:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
354:
350:
337:
336:states that
327:
317:
313:
305:
301:
297:
293:
287:
282:
278:
277:of a subset
272:
267:
265:
118:. Formally,
115:
111:
107:
103:
99:
95:
87:
83:
79:
75:
71:
67:
63:
57:
35:
32:
26:
2744:Elias Stein
1333:, we have:
1113:varies in {
567:strong-type
330:G. H. Hardy
29:mathematics
3251:Categories
3140:What's New
3119:References
2788:uncentered
2724:Discussion
2035:. See the
1461:= 5. Then
1013:such that
990:such that
986:For every
310:measurable
290:continuous
54:Definition
3032:∫
2987:∈
2959:Δ
2876:∫
2831:∈
2806:∗
2694:→
2650:→
2641:‖
2634:−
2621:‖
2565:→
2530:→
2521:‖
2514:‖
2485:‖
2478:‖
2473:ε
2459:≤
2448:ε
2439:Ω
2394:Ω
2385:Ω
2376:Ω
2373:≤
2367:Ω
2328:We write
2271:∫
2161:→
2150:−
2123:→
2097:Ω
1919:‖
1912:‖
1866:−
1826:∫
1820:∞
1811:∫
1736:∫
1707:−
1694:∞
1685:∫
1678:≤
1664:‖
1654:‖
1589:−
1576:∞
1567:∫
1540:−
1503:∫
1499:∫
1482:‖
1472:‖
1391:∫
1372:≤
1256:∫
1236:≤
1207:∑
1193:≤
1025:∫
939:∈
932:⋃
928:⊂
913:∈
906:⋃
691:‖
684:‖
665:≤
631:‖
621:‖
515:‖
508:‖
503:λ
477:λ
201:∫
62:function
3107:See also
1325:)| >
1146:covers {
948:′
881:′
379:is weak
66: :
42:used in
40:operator
2027:of the
998:) >
974:where 5
415:) >
342:bounded
266:where |
3145:22 May
3088:where
2941:dyadic
2340:where
2200:where
2086:) and
1317:) if |
1101:Thus {
811:Lemma.
31:, the
1457:with
1150:>
1117:>
1105:>
786:Proof
399:>
348:from
344:as a
3147:2011
2735:and
2445:>
1956:and
1843:>
1753:>
1611:>
1408:>
1358:>
1309:) =
1179:>
1060:>
813:Let
580:For
488:<
474:>
426:For
332:and
296:and
155:>
46:and
3174:doi
2980:sup
2824:sup
2761:p,d
2754:p,d
2732:p,d
2558:lim
2416:≤ 2
1301:by
978:is
767:p,d
745:p,d
607:p,d
600:),
340:is
292:in
148:sup
114:at
94:of
50:.
27:In
3253::
3239:,
3224:73
3211:,
3193:,
3164:.
3138:.
3126:^
2716:→
2418:Mg
2348:∈
2336:+
2332:=
2078:∈
1148:Mf
1115:Mf
1103:Mf
992:Mf
770:=
573::
407:|
387:∈
385:Mf
377:Mf
367:∈
320:.
314:Mf
302:Mf
285:.
281:⊂
106:,
88:Mf
82:∈
76:Mf
70:→
3182:.
3176::
3170:7
3149:.
3101:x
3092:x
3090:Q
3073:y
3070:d
3066:|
3062:)
3059:y
3056:(
3053:f
3049:|
3041:x
3037:Q
3024:|
3018:x
3014:Q
3009:|
3004:1
2995:x
2991:Q
2984:x
2976:=
2973:)
2970:x
2967:(
2964:f
2955:M
2936:x
2934:B
2917:y
2914:d
2910:|
2906:)
2903:y
2900:(
2897:f
2893:|
2885:x
2881:B
2868:|
2862:x
2858:B
2853:|
2848:1
2839:x
2835:B
2828:x
2820:=
2817:)
2814:x
2811:(
2802:f
2780:p
2775:p
2773:C
2768:p
2766:C
2759:C
2752:C
2748:p
2739:d
2737:C
2730:C
2718:f
2713:r
2711:f
2697:f
2687:k
2683:r
2678:f
2665:(
2653:0
2645:1
2637:f
2629:r
2625:f
2611:x
2609:(
2607:f
2603:x
2589:)
2586:x
2583:(
2578:r
2574:f
2568:0
2562:r
2547:f
2533:0
2525:1
2517:g
2489:1
2481:g
2469:M
2465:2
2455:|
2451:}
2442:g
2436:{
2432:|
2414:g
2397:g
2391:=
2388:h
2382:+
2379:g
2370:f
2354:R
2352:(
2350:L
2346:g
2342:h
2338:g
2334:h
2330:f
2313:.
2310:y
2307:d
2304:)
2301:y
2298:(
2295:f
2290:)
2287:r
2284:,
2281:x
2278:(
2275:B
2263:|
2259:)
2256:r
2253:,
2250:x
2247:(
2244:B
2240:|
2235:1
2230:=
2227:)
2224:x
2221:(
2216:r
2212:f
2185:)
2182:x
2179:(
2174:r
2170:f
2164:0
2158:r
2147:)
2144:x
2141:(
2136:r
2132:f
2126:0
2120:r
2112:=
2109:)
2106:x
2103:(
2100:f
2084:R
2082:(
2080:L
2076:f
2009:d
2005:3
1982:d
1978:5
1974:=
1971:C
1958:d
1954:p
1949:p
1947:C
1928:p
1923:p
1915:f
1907:p
1903:C
1899:=
1896:t
1893:d
1890:x
1887:d
1883:|
1879:f
1875:|
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