1033:
469:
1948:
2184:
2452:), a pseudo-differential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudo-differential operators to a sequence of algebraic problems involving their symbols, and this is the essence of
1545:
1732:
682:
2468:. The singularity of the kernel on the diagonal depends on the degree of the corresponding operator. In fact, if the symbol satisfies the above differential inequalities with m †0, it can be shown that the kernel is a
1408:
844:
284:
297:
1160:
174:
1307:
829:
543:
1790:
2046:
2252:
2369:, and its symbol can be calculated. This means that one can solve linear elliptic differential equations more or less explicitly by using the theory of pseudo-differential operators.
2313:
2002:
1419:
1596:
574:
1219:
2432:
582:
2376:
in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator. Pseudo-differential operators are
718:
197:
2761:
1028:{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int e^{ix\xi }{\hat {u}}(\xi )d\xi ={\frac {1}{(2\pi )^{n}}}\iint e^{i(x-y)\xi }u(y)\,dy\,d\xi }
1330:
464:{\displaystyle \quad P(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{i(x-y)\xi }P(\xi )u(y)\,dy\,d\xi }
223:
1056:
2675:
2657:
2647:
2634:
2624:
116:
2642:, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981.
1943:{\displaystyle \quad P(x,D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }P(x,\xi ){\hat {u}}(\xi )\,d\xi }
1234:
752:
82:
2736:
490:
1755:
Here we view pseudo-differential operators as a generalization of differential operators. We extend formula (1) as follows. A
2756:
2358:
43:
2179:{\displaystyle |\partial _{\xi }^{\alpha }\partial _{x}^{\beta }P(x,\xi )|\leq C_{\alpha ,\beta }\,(1+|\xi |)^{m-|\alpha |}}
835:
2719:
2497:
2746:
2741:
2714:
2322:
Linear differential operators of order m with smooth bounded coefficients are pseudo-differential operators of order
2502:
2751:
1584:
51:
2709:
2492:
2484:
for a definition of pseudo-differential operators in the context of differential algebras and differential rings.
2215:
2652:
F. G. Friedlander and M. Joshi, Introduction to the Theory of
Distributions, Cambridge University Press 1999.
2381:
1540:{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int e^{ix\xi }{\frac {1}{P(\xi )}}{\hat {f}}(\xi )\,d\xi .}
107:
39:
31:
2277:
1969:
2507:
2481:
1727:{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\iint e^{i(x-y)\xi }{\frac {1}{P(\xi )}}f(y)\,dy\,d\xi .}
86:
55:
47:
2453:
2592:
2465:
2350:. The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator.
677:{\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\cdots (-i\partial _{n})^{\alpha _{n}}}
552:
213:
1185:
2671:
2663:
2653:
2643:
2630:
2620:
2616:
2487:
2469:
2401:
2005:
204:
90:
75:
2613:. Pseudo-Differential Operators. Theory and Applications, 3. BirkhÀuser Verlag, Basel, 2010.
2584:
17:
2639:
739:
71:
2668:
The
Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators
700:
2572:
2568:
182:
67:
2629:
M. A. Shubin, Pseudodifferential
Operators and Spectral Theory, Springer-Verlag 2001.
2384:
they do not create a singularity at points where the distribution was already smooth.
2730:
2702:
2365:) and invertible, then its inverse is a pseudo-differential operator of order −
66:
The study of pseudo-differential operators began in the mid 1960s with the work of
2688:. Lecture Notes Series, 46. Aarhus Universitet, Matematisk Institut, Aarhus, 1976.
2255:
2560:
Harmonic
Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals
2555:
546:
93:
for assistance with understanding the theory of pseudo-differential operators.
2438:
1747:(ξ) is not a polynomial function, but a function of a more general kind.
2611:
Metrics on the phase space and non-selfadjoint pseudo-differential operators
1590:
In the last formula, write out the
Fourier transform of ƒ to obtain
2698:
2596:
1224:
we (formally) apply the
Fourier transform on both sides and obtain the
42:. Pseudo-differential operators are used extensively in the theory of
1403:{\displaystyle {\hat {u}}(\xi )={\frac {1}{P(\xi )}}{\hat {f}}(\xi )}
2588:
279:{\displaystyle P(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha },}
1155:{\displaystyle P(D_{x})\,e^{i(x-y)\xi }=e^{i(x-y)\xi }\,P(\xi )}
2686:
Pseudo-differential operators and applications: an introduction
2387:
Just as a differential operator can be expressed in terms of
1175:
Representation of solutions to partial differential equations
169:{\displaystyle P(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}
2619:, Pseudodifferential Operators, Princeton Univ. Press 1981.
1583:
The last assumption can be weakened by using the theory of
1302:{\displaystyle P(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi ).}
81:
They played an influential role in the second proof of the
2338:
is again a pseudo-differential operator and the symbol of
824:{\displaystyle {\hat {u}}(\xi ):=\int e^{-iy\xi }u(y)\,dy}
538:{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}
1587:. The first two assumptions can be weakened as follows.
102:
Linear differential operators with constant coefficients
50:, e.g. in mathematical models that include ultrametric
2219:
2404:
2280:
2218:
2049:
2032:,ξ) is an infinitely differentiable function on
1972:
1793:
1599:
1422:
1333:
1316:(ξ) is never zero when ξ ∈
1237:
1188:
1059:
847:
755:
720:
to facilitate the calculation of
Fourier transforms.
703:
585:
555:
493:
300:
226:
203:. This operator can be written as a composition of a
185:
119:
2464:
Pseudo-differential operators can be represented by
2426:
2307:
2246:
2178:
1996:
1942:
1726:
1539:
1402:
1301:
1213:
1154:
1027:
823:
712:
676:
568:
537:
463:
278:
191:
168:
2197:, all multiindices α,β, some constants
1579:and ƒ have a well defined Fourier transform.
2380:, which means informally that when applied to a
2020:,ξ) in the integrand belongs to a certain
289:and an inverse Fourier transform, in the form:
1413:By Fourier's inversion formula, a solution is
2575:(1968), "The Index of Elliptic Operators I",
8:
2538:
1774:is an operator whose value on the function
1751:Definition of pseudo-differential operators
1179:To solve the partial differential equation
734:The Fourier transform of a smooth function
2342:can be calculated by using the symbols of
2247:{\displaystyle \scriptstyle {S_{1,0}^{m}}}
697:-th variable. We introduce the constants
693:means differentiation with respect to the
687:is an iterated partial derivative, where â
2699:Lectures on Pseudo-differential Operators
2423:
2403:
2296:
2285:
2279:
2236:
2225:
2220:
2217:
2169:
2161:
2154:
2145:
2137:
2127:
2115:
2103:
2079:
2074:
2064:
2059:
2050:
2048:
1974:
1973:
1971:
1933:
1913:
1912:
1879:
1867:
1863:
1862:
1860:
1847:
1828:
1792:
1714:
1707:
1674:
1650:
1634:
1615:
1598:
1561:) is a linear differential operator with
1527:
1507:
1506:
1485:
1473:
1457:
1438:
1421:
1380:
1379:
1358:
1335:
1334:
1332:
1276:
1275:
1252:
1251:
1250:
1236:
1201:
1187:
1139:
1115:
1084:
1079:
1070:
1058:
1018:
1011:
975:
959:
940:
911:
910:
898:
882:
863:
846:
814:
787:
757:
756:
754:
702:
666:
661:
651:
627:
622:
612:
590:
584:
560:
554:
526:
507:
492:
454:
447:
399:
387:
383:
382:
380:
368:
364:
363:
361:
348:
329:
299:
267:
262:
256:
246:
225:
184:
160:
155:
149:
139:
118:
2519:
2272:pseudo-differential operator of order m
211:by the polynomial function (called the
2460:Kernel of pseudo-differential operator
2526:
2330:of two pseudo-differential operators
7:
2353:If a differential operator of order
1784:
291:
1320:, then it is possible to divide by
2282:
2071:
2056:
648:
609:
38:is an extension of the concept of
25:
2308:{\displaystyle \Psi _{1,0}^{m}.}
1997:{\displaystyle {\hat {u}}(\xi )}
1794:
301:
179:which acts on smooth functions
2762:Partial differential equations
2710:"Pseudo-differential operator"
2420:
2408:
2170:
2162:
2151:
2146:
2138:
2128:
2104:
2100:
2088:
2051:
1991:
1985:
1979:
1930:
1924:
1918:
1909:
1897:
1844:
1834:
1822:
1816:
1810:
1798:
1704:
1698:
1689:
1683:
1666:
1654:
1631:
1621:
1609:
1603:
1524:
1518:
1512:
1500:
1494:
1454:
1444:
1432:
1426:
1397:
1391:
1385:
1373:
1367:
1352:
1346:
1340:
1293:
1287:
1281:
1269:
1263:
1257:
1247:
1241:
1198:
1192:
1149:
1143:
1131:
1119:
1100:
1088:
1076:
1063:
1008:
1002:
991:
979:
956:
946:
928:
922:
916:
879:
869:
857:
851:
811:
805:
774:
768:
762:
658:
638:
619:
599:
532:
500:
444:
438:
432:
426:
415:
403:
345:
335:
323:
317:
311:
305:
236:
230:
129:
123:
44:partial differential equations
1:
2498:Oscillatory integral operator
2258:. The corresponding operator
89:. Atiyah and Singer thanked
52:pseudo-differential equations
27:Type of differential operator
2562:, Princeton University Press
2212:belongs to the symbol class
1757:pseudo-differential operator
1737:This is similar to formula (
1046:) to this representation of
110:with constant coefficients,
36:pseudo-differential operator
2715:Encyclopedia of Mathematics
2372:Differential operators are
1739:
1167:
836:Fourier's inversion formula
726:
569:{\displaystyle a_{\alpha }}
83:AtiyahâSinger index theorem
78:, Unterberger and Bokobza.
18:Pseudodifferential operator
2778:
2503:Sato's fundamental theorem
2493:Fourier integral operator
2274:and belongs to the class
1550:Here it is assumed that:
1214:{\displaystyle P(D)\,u=f}
576:are complex numbers, and
2539:Atiyah & Singer 1968
2470:singular integral kernel
2427:{\displaystyle p(x,D)\,}
2391: = −id/d
199:with compact support in
1572:(ξ) is never zero,
724:Derivation of formula (
2737:Differential operators
2428:
2309:
2248:
2180:
1998:
1944:
1728:
1541:
1404:
1303:
1215:
1156:
1029:
825:
714:
678:
570:
539:
465:
280:
193:
170:
2757:Generalized functions
2577:Annals of Mathematics
2448:(which is called the
2429:
2310:
2249:
2204:and some real number
2181:
1999:
1945:
1729:
1542:
1405:
1304:
1216:
1165:one obtains formula (
1157:
1030:
826:
715:
679:
571:
540:
466:
281:
194:
171:
108:differential operator
40:differential operator
32:mathematical analysis
2508:Operational calculus
2482:Differential algebra
2402:
2359:(uniformly) elliptic
2278:
2216:
2193:,ξ ∈
2047:
1970:
1791:
1597:
1420:
1331:
1235:
1186:
1057:
845:
753:
701:
583:
553:
491:
298:
224:
183:
117:
48:quantum field theory
2747:Functional analysis
2742:Microlocal analysis
2684:André Unterberger,
2454:microlocal analysis
2301:
2241:
2084:
2069:
2024:. For instance, if
1778:is the function of
740:compactly supported
2573:Singer, Isadore M.
2569:Atiyah, Michael F.
2424:
2326:. The composition
2305:
2281:
2244:
2243:
2221:
2176:
2070:
2055:
2040:with the property
2036: ×
1994:
1940:
1724:
1537:
1400:
1299:
1211:
1152:
1025:
821:
713:{\displaystyle -i}
710:
674:
566:
535:
461:
276:
251:
189:
166:
144:
106:Consider a linear
2752:Harmonic analysis
2617:Michael E. Taylor
2609:Nicolas Lerner,
2488:Fourier transform
2006:Fourier transform
1982:
1964:
1963:
1921:
1854:
1743:), except that 1/
1693:
1641:
1515:
1504:
1464:
1388:
1377:
1343:
1284:
1260:
966:
919:
889:
765:
485:
484:
355:
242:
205:Fourier transform
192:{\displaystyle u}
135:
16:(Redirected from
2769:
2723:
2681:
2599:
2563:
2542:
2536:
2530:
2524:
2433:
2431:
2430:
2425:
2314:
2312:
2311:
2306:
2300:
2295:
2253:
2251:
2250:
2245:
2242:
2240:
2235:
2185:
2183:
2182:
2177:
2175:
2174:
2173:
2165:
2149:
2141:
2126:
2125:
2107:
2083:
2078:
2068:
2063:
2054:
2003:
2001:
2000:
1995:
1984:
1983:
1975:
1958:
1949:
1947:
1946:
1941:
1923:
1922:
1914:
1893:
1892:
1874:
1873:
1872:
1871:
1866:
1855:
1853:
1852:
1851:
1829:
1785:
1733:
1731:
1730:
1725:
1694:
1692:
1675:
1673:
1672:
1642:
1640:
1639:
1638:
1616:
1546:
1544:
1543:
1538:
1517:
1516:
1508:
1505:
1503:
1486:
1484:
1483:
1465:
1463:
1462:
1461:
1439:
1409:
1407:
1406:
1401:
1390:
1389:
1381:
1378:
1376:
1359:
1345:
1344:
1336:
1308:
1306:
1305:
1300:
1286:
1285:
1277:
1262:
1261:
1253:
1220:
1218:
1217:
1212:
1161:
1159:
1158:
1153:
1138:
1137:
1107:
1106:
1075:
1074:
1034:
1032:
1031:
1026:
998:
997:
967:
965:
964:
963:
941:
921:
920:
912:
909:
908:
890:
888:
887:
886:
864:
830:
828:
827:
822:
801:
800:
767:
766:
758:
719:
717:
716:
711:
683:
681:
680:
675:
673:
672:
671:
670:
656:
655:
634:
633:
632:
631:
617:
616:
595:
594:
575:
573:
572:
567:
565:
564:
544:
542:
541:
536:
531:
530:
512:
511:
479:
470:
468:
467:
462:
422:
421:
394:
393:
392:
391:
386:
375:
374:
373:
372:
367:
356:
354:
353:
352:
330:
292:
285:
283:
282:
277:
272:
271:
261:
260:
250:
198:
196:
195:
190:
175:
173:
172:
167:
165:
164:
154:
153:
143:
21:
2777:
2776:
2772:
2771:
2770:
2768:
2767:
2766:
2727:
2726:
2708:
2695:
2678:
2664:Hörmander, Lars
2662:
2640:Francois Treves
2606:
2604:Further reading
2589:10.2307/1970715
2567:
2554:
2551:
2546:
2545:
2537:
2533:
2525:
2521:
2516:
2478:
2462:
2400:
2399:
2320:
2276:
2275:
2214:
2213:
2203:
2150:
2111:
2045:
2044:
2012:and the symbol
1968:
1967:
1956:
1875:
1861:
1856:
1843:
1833:
1789:
1788:
1753:
1679:
1646:
1630:
1620:
1595:
1594:
1490:
1469:
1453:
1443:
1418:
1417:
1363:
1329:
1328:
1233:
1232:
1184:
1183:
1177:
1111:
1080:
1066:
1055:
1054:
971:
955:
945:
894:
878:
868:
843:
842:
783:
751:
750:
699:
698:
692:
662:
657:
647:
623:
618:
608:
586:
581:
580:
556:
551:
550:
522:
503:
489:
488:
477:
395:
381:
376:
362:
357:
344:
334:
296:
295:
263:
252:
222:
221:
181:
180:
156:
145:
115:
114:
104:
99:
64:
56:non-Archimedean
28:
23:
22:
15:
12:
11:
5:
2775:
2773:
2765:
2764:
2759:
2754:
2749:
2744:
2739:
2729:
2728:
2725:
2724:
2706:
2694:
2693:External links
2691:
2690:
2689:
2682:
2676:
2660:
2650:
2637:
2627:
2614:
2605:
2602:
2601:
2600:
2583:(3): 484â530,
2565:
2550:
2547:
2544:
2543:
2531:
2518:
2517:
2515:
2512:
2511:
2510:
2505:
2500:
2495:
2490:
2485:
2477:
2474:
2461:
2458:
2435:
2434:
2422:
2419:
2416:
2413:
2410:
2407:
2319:
2316:
2304:
2299:
2294:
2291:
2288:
2284:
2270:) is called a
2239:
2234:
2231:
2228:
2224:
2202:α, β
2201:
2187:
2186:
2172:
2168:
2164:
2160:
2157:
2153:
2148:
2144:
2140:
2136:
2133:
2130:
2124:
2121:
2118:
2114:
2110:
2106:
2102:
2099:
2096:
2093:
2090:
2087:
2082:
2077:
2073:
2067:
2062:
2058:
2053:
1993:
1990:
1987:
1981:
1978:
1962:
1961:
1952:
1950:
1939:
1936:
1932:
1929:
1926:
1920:
1917:
1911:
1908:
1905:
1902:
1899:
1896:
1891:
1888:
1885:
1882:
1878:
1870:
1865:
1859:
1850:
1846:
1842:
1839:
1836:
1832:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1752:
1749:
1735:
1734:
1723:
1720:
1717:
1713:
1710:
1706:
1703:
1700:
1697:
1691:
1688:
1685:
1682:
1678:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1649:
1645:
1637:
1633:
1629:
1626:
1623:
1619:
1614:
1611:
1608:
1605:
1602:
1581:
1580:
1573:
1566:
1548:
1547:
1536:
1533:
1530:
1526:
1523:
1520:
1514:
1511:
1502:
1499:
1496:
1493:
1489:
1482:
1479:
1476:
1472:
1468:
1460:
1456:
1452:
1449:
1446:
1442:
1437:
1434:
1431:
1428:
1425:
1411:
1410:
1399:
1396:
1393:
1387:
1384:
1375:
1372:
1369:
1366:
1362:
1357:
1354:
1351:
1348:
1342:
1339:
1312:If the symbol
1310:
1309:
1298:
1295:
1292:
1289:
1283:
1280:
1274:
1271:
1268:
1265:
1259:
1256:
1249:
1246:
1243:
1240:
1222:
1221:
1210:
1207:
1204:
1200:
1197:
1194:
1191:
1176:
1173:
1163:
1162:
1151:
1148:
1145:
1142:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1114:
1110:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1083:
1078:
1073:
1069:
1065:
1062:
1036:
1035:
1024:
1021:
1017:
1014:
1010:
1007:
1004:
1001:
996:
993:
990:
987:
984:
981:
978:
974:
970:
962:
958:
954:
951:
948:
944:
939:
936:
933:
930:
927:
924:
918:
915:
907:
904:
901:
897:
893:
885:
881:
877:
874:
871:
867:
862:
859:
856:
853:
850:
832:
831:
820:
817:
813:
810:
807:
804:
799:
796:
793:
790:
786:
782:
779:
776:
773:
770:
764:
761:
732:
731:
709:
706:
688:
685:
684:
669:
665:
660:
654:
650:
646:
643:
640:
637:
630:
626:
621:
615:
611:
607:
604:
601:
598:
593:
589:
563:
559:
534:
529:
525:
521:
518:
515:
510:
506:
502:
499:
496:
483:
482:
473:
471:
460:
457:
453:
450:
446:
443:
440:
437:
434:
431:
428:
425:
420:
417:
414:
411:
408:
405:
402:
398:
390:
385:
379:
371:
366:
360:
351:
347:
343:
340:
337:
333:
328:
325:
322:
319:
316:
313:
310:
307:
304:
287:
286:
275:
270:
266:
259:
255:
249:
245:
241:
238:
235:
232:
229:
209:multiplication
188:
177:
176:
163:
159:
152:
148:
142:
138:
134:
131:
128:
125:
122:
103:
100:
98:
95:
63:
60:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2774:
2763:
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2734:
2732:
2721:
2717:
2716:
2711:
2707:
2705:on arxiv.org.
2704:
2703:Mark S. Joshi
2700:
2697:
2696:
2692:
2687:
2683:
2679:
2677:3-540-49937-7
2673:
2669:
2665:
2661:
2659:
2658:0-521-64971-4
2655:
2651:
2649:
2648:0-306-40404-4
2645:
2641:
2638:
2636:
2635:3-540-41195-X
2632:
2628:
2626:
2625:0-691-08282-0
2622:
2618:
2615:
2612:
2608:
2607:
2603:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2566:
2561:
2557:
2553:
2552:
2548:
2541:, p. 486
2540:
2535:
2532:
2528:
2523:
2520:
2513:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2483:
2480:
2479:
2475:
2473:
2471:
2467:
2459:
2457:
2455:
2451:
2447:
2443:
2440:
2417:
2414:
2411:
2405:
2398:
2397:
2396:
2394:
2390:
2385:
2383:
2379:
2375:
2370:
2368:
2364:
2360:
2356:
2351:
2349:
2345:
2341:
2337:
2333:
2329:
2325:
2317:
2315:
2302:
2297:
2292:
2289:
2286:
2273:
2269:
2265:
2261:
2257:
2237:
2232:
2229:
2226:
2222:
2211:
2207:
2200:
2196:
2192:
2166:
2158:
2155:
2142:
2134:
2131:
2122:
2119:
2116:
2112:
2108:
2097:
2094:
2091:
2085:
2080:
2075:
2065:
2060:
2043:
2042:
2041:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
1988:
1976:
1960:
1953:
1951:
1937:
1934:
1927:
1915:
1906:
1903:
1900:
1894:
1889:
1886:
1883:
1880:
1876:
1868:
1857:
1848:
1840:
1837:
1830:
1825:
1819:
1813:
1807:
1804:
1801:
1795:
1787:
1786:
1783:
1781:
1777:
1773:
1769:
1765:
1761:
1758:
1750:
1748:
1746:
1742:
1741:
1721:
1718:
1715:
1711:
1708:
1701:
1695:
1686:
1680:
1676:
1669:
1663:
1660:
1657:
1651:
1647:
1643:
1635:
1627:
1624:
1617:
1612:
1606:
1600:
1593:
1592:
1591:
1588:
1586:
1585:distributions
1578:
1574:
1571:
1567:
1565:coefficients,
1564:
1560:
1556:
1553:
1552:
1551:
1534:
1531:
1528:
1521:
1509:
1497:
1491:
1487:
1480:
1477:
1474:
1470:
1466:
1458:
1450:
1447:
1440:
1435:
1429:
1423:
1416:
1415:
1414:
1394:
1382:
1370:
1364:
1360:
1355:
1349:
1337:
1327:
1326:
1325:
1323:
1319:
1315:
1296:
1290:
1278:
1272:
1266:
1254:
1244:
1238:
1231:
1230:
1229:
1227:
1208:
1205:
1202:
1195:
1189:
1182:
1181:
1180:
1174:
1172:
1170:
1169:
1146:
1140:
1134:
1128:
1125:
1122:
1116:
1112:
1108:
1103:
1097:
1094:
1091:
1085:
1081:
1071:
1067:
1060:
1053:
1052:
1051:
1049:
1045:
1041:
1022:
1019:
1015:
1012:
1005:
999:
994:
988:
985:
982:
976:
972:
968:
960:
952:
949:
942:
937:
934:
931:
925:
913:
905:
902:
899:
895:
891:
883:
875:
872:
865:
860:
854:
848:
841:
840:
839:
837:
818:
815:
808:
802:
797:
794:
791:
788:
784:
780:
777:
771:
759:
749:
748:
747:
745:
741:
737:
729:
728:
723:
722:
721:
707:
704:
696:
691:
667:
663:
652:
644:
641:
635:
628:
624:
613:
605:
602:
596:
591:
587:
579:
578:
577:
561:
557:
548:
527:
523:
519:
516:
513:
508:
504:
497:
494:
481:
474:
472:
458:
455:
451:
448:
441:
435:
429:
423:
418:
412:
409:
406:
400:
396:
388:
377:
369:
358:
349:
341:
338:
331:
326:
320:
314:
308:
302:
294:
293:
290:
273:
268:
264:
257:
253:
247:
243:
239:
233:
227:
220:
219:
218:
216:
215:
210:
206:
202:
186:
161:
157:
150:
146:
140:
136:
132:
126:
120:
113:
112:
111:
109:
101:
96:
94:
92:
88:
84:
79:
77:
73:
69:
61:
59:
57:
53:
49:
45:
41:
37:
33:
19:
2713:
2685:
2670:. Springer.
2667:
2610:
2580:
2576:
2559:
2556:Stein, Elias
2534:
2522:
2463:
2449:
2445:
2441:
2436:
2395:in the form
2392:
2388:
2386:
2382:distribution
2378:pseudo-local
2377:
2373:
2371:
2366:
2362:
2354:
2352:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2321:
2271:
2267:
2263:
2259:
2209:
2205:
2198:
2194:
2190:
2188:
2037:
2033:
2029:
2025:
2022:symbol class
2021:
2017:
2013:
2009:
1965:
1954:
1779:
1775:
1771:
1767:
1763:
1759:
1756:
1754:
1744:
1738:
1736:
1589:
1582:
1576:
1569:
1562:
1558:
1554:
1549:
1412:
1321:
1317:
1313:
1311:
1225:
1223:
1178:
1166:
1164:
1047:
1043:
1039:
1038:By applying
1037:
833:
743:
735:
733:
725:
694:
689:
686:
486:
475:
288:
212:
208:
200:
178:
105:
80:
65:
35:
29:
2529:, Chapter 6
1568:its symbol
547:multi-index
207:, a simple
2731:Categories
2549:References
2527:Stein 1993
2439:polynomial
2361:(of order
2318:Properties
1324:(ξ):
1050:and using
97:Motivation
2720:EMS Press
2514:Footnotes
2283:Ψ
2256:Hörmander
2167:α
2159:−
2143:ξ
2123:β
2117:α
2109:≤
2098:ξ
2081:β
2072:∂
2066:α
2061:ξ
2057:∂
1989:ξ
1980:^
1938:ξ
1928:ξ
1919:^
1907:ξ
1890:ξ
1887:⋅
1858:∫
1841:π
1719:ξ
1687:ξ
1670:ξ
1661:−
1644:∬
1628:π
1532:ξ
1522:ξ
1513:^
1498:ξ
1481:ξ
1467:∫
1451:π
1395:ξ
1386:^
1371:ξ
1350:ξ
1341:^
1291:ξ
1282:^
1267:ξ
1258:^
1245:ξ
1228:equation
1226:algebraic
1147:ξ
1135:ξ
1126:−
1104:ξ
1095:−
1023:ξ
995:ξ
986:−
969:∬
953:π
935:ξ
926:ξ
917:^
906:ξ
892:∫
876:π
798:ξ
789:−
781:∫
772:ξ
763:^
705:−
664:α
649:∂
642:−
636:⋯
625:α
610:∂
603:−
592:α
562:α
524:α
517:…
505:α
495:α
459:ξ
430:ξ
419:ξ
410:−
378:∫
359:∫
342:π
269:α
265:ξ
258:α
248:α
244:∑
234:ξ
162:α
151:α
141:α
137:∑
91:Hörmander
76:Hörmander
72:Nirenberg
2666:(1987).
2558:(1993),
2476:See also
2189:for all
1563:constant
87:K-theory
2722:, 2001
2597:1970715
2466:kernels
2334:,
2208:, then
2004:is the
62:History
58:space.
2674:
2656:
2646:
2633:
2623:
2595:
2450:symbol
2437:for a
1966:where
838:gives
487:Here,
214:symbol
2593:JSTOR
2374:local
1770:) on
1575:both
746:, is
545:is a
54:in a
2672:ISBN
2654:ISBN
2644:ISBN
2631:ISBN
2621:ISBN
2346:and
1776:u(x)
834:and
85:via
68:Kohn
46:and
2701:by
2585:doi
2472:.
2444:in
2357:is
2254:of
2008:of
1171:).
742:in
30:In
2733::
2718:,
2712:,
2591:,
2581:87
2579:,
2571:;
2456:.
2340:PQ
2328:PQ
1782::
778::=
738:,
549:,
217:)
133::=
74:,
70:,
34:a
2680:.
2587::
2564:.
2446:D
2442:p
2421:)
2418:D
2415:,
2412:x
2409:(
2406:p
2393:x
2389:D
2367:m
2363:m
2355:m
2348:Q
2344:P
2336:Q
2332:P
2324:m
2303:.
2298:m
2293:0
2290:,
2287:1
2268:D
2266:,
2264:x
2262:(
2260:P
2238:m
2233:0
2230:,
2227:1
2223:S
2210:P
2206:m
2199:C
2195:R
2191:x
2171:|
2163:|
2156:m
2152:)
2147:|
2139:|
2135:+
2132:1
2129:(
2120:,
2113:C
2105:|
2101:)
2095:,
2092:x
2089:(
2086:P
2076:x
2052:|
2038:R
2034:R
2030:x
2028:(
2026:P
2018:x
2016:(
2014:P
2010:u
1992:)
1986:(
1977:u
1959:)
1957:2
1955:(
1935:d
1931:)
1925:(
1916:u
1910:)
1904:,
1901:x
1898:(
1895:P
1884:x
1881:i
1877:e
1869:n
1864:R
1849:n
1845:)
1838:2
1835:(
1831:1
1826:=
1823:)
1820:x
1817:(
1814:u
1811:)
1808:D
1805:,
1802:x
1799:(
1796:P
1780:x
1772:R
1768:D
1766:,
1764:x
1762:(
1760:P
1745:P
1740:1
1722:.
1716:d
1712:y
1709:d
1705:)
1702:y
1699:(
1696:f
1690:)
1684:(
1681:P
1677:1
1667:)
1664:y
1658:x
1655:(
1652:i
1648:e
1636:n
1632:)
1625:2
1622:(
1618:1
1613:=
1610:)
1607:x
1604:(
1601:u
1577:u
1570:P
1559:D
1557:(
1555:P
1535:.
1529:d
1525:)
1519:(
1510:f
1501:)
1495:(
1492:P
1488:1
1478:x
1475:i
1471:e
1459:n
1455:)
1448:2
1445:(
1441:1
1436:=
1433:)
1430:x
1427:(
1424:u
1398:)
1392:(
1383:f
1374:)
1368:(
1365:P
1361:1
1356:=
1353:)
1347:(
1338:u
1322:P
1318:R
1314:P
1297:.
1294:)
1288:(
1279:f
1273:=
1270:)
1264:(
1255:u
1248:)
1242:(
1239:P
1209:f
1206:=
1203:u
1199:)
1196:D
1193:(
1190:P
1168:1
1150:)
1144:(
1141:P
1132:)
1129:y
1123:x
1120:(
1117:i
1113:e
1109:=
1101:)
1098:y
1092:x
1089:(
1086:i
1082:e
1077:)
1072:x
1068:D
1064:(
1061:P
1048:u
1044:D
1042:(
1040:P
1020:d
1016:y
1013:d
1009:)
1006:y
1003:(
1000:u
992:)
989:y
983:x
980:(
977:i
973:e
961:n
957:)
950:2
947:(
943:1
938:=
932:d
929:)
923:(
914:u
903:x
900:i
896:e
884:n
880:)
873:2
870:(
866:1
861:=
858:)
855:x
852:(
849:u
819:y
816:d
812:)
809:y
806:(
803:u
795:y
792:i
785:e
775:)
769:(
760:u
744:R
736:u
730:)
727:1
708:i
695:j
690:j
668:n
659:)
653:n
645:i
639:(
629:1
620:)
614:1
606:i
600:(
597:=
588:D
558:a
533:)
528:n
520:,
514:,
509:1
501:(
498:=
480:)
478:1
476:(
456:d
452:y
449:d
445:)
442:y
439:(
436:u
433:)
427:(
424:P
416:)
413:y
407:x
404:(
401:i
397:e
389:n
384:R
370:n
365:R
350:n
346:)
339:2
336:(
332:1
327:=
324:)
321:x
318:(
315:u
312:)
309:D
306:(
303:P
274:,
254:a
240:=
237:)
231:(
228:P
201:R
187:u
158:D
147:a
130:)
127:D
124:(
121:P
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.