Acta mathematica scientia, Series B >
Lp CONTINUITY OF HÖRMANDER SYMBOL OPERATORS OpSm0,0 AND NUMERICAL ALGORITHM
Received date: 2008-12-29
Online published: 2011-07-20
Supported by
Supported by the Doctoral programme foundation of National Education Ministry of China.
If we use Littlewood-Paley decomposition, there is no pseudo-orthogonality for H¨ormander symbol operators OpSm0,0, which is different to the case Smρ,δ (0 ≤δ < ρ ≤ 1). In this paper, we use a special numerical algorithm based on wavelets to study the Lp
continuity of non infinite smooth operators OpSm0,0; in fact, we apply first special wavelets to symbol to get special basic operators, then we regroup all the special basic operators at given scale and prove that such scale operator’s continuity decreases very fast, we sum such scale operators and a symbol operator can be approached by very good compact operators. By correlation of basic operators, we get very exact pseudo-orthogonality and also L2→L2 continuity for scale operators. By considering the influence region of scale operator, we get H1(= F0,21 ) → L1 continuity and L1 →BMO continuity. By interpolation theorem, we get also Lp(= F0,2p ) → Lp continuity for 1 < p < ∞. Our results are sharp for F0,2p →Lp continuity when 1 ≤ p ≤ 2, that is to say, we find out the exact order of derivations for
which the symbols can ensure the resulting operators to be bounded on these spaces.
YANG Qi-Xiang . Lp CONTINUITY OF HÖRMANDER SYMBOL OPERATORS OpSm0,0 AND NUMERICAL ALGORITHM[J]. Acta mathematica scientia, Series B, 2011 , 31(4) : 1517 -1534 . DOI: 10.1016/S0252-9602(11)60338-X
[1] Bourdaud G. Lp-estimates for certain non-regular pseudo-differential operators. Comm Partial Differ Equ, 1982, 7(9): 1023–1033
[2] Bourdaud G, Meyer Y. In´egalit´es L2 pr´ecis´ees pour la classe S0 0,0. Bull Soc Math France, 1988, 116:401–s412
[3] Boulkhemair A. L2 estimates for pseudodifferential operators. Ann Scuola Norm Sup Pisa Cl Sci, 1995,22(4): 155–183
[4] Calder´on A P, Vaillancourt R. A Class of bounded pseudo-differential operators. Proc Nat Acad Sci, 1972,69: 1185–1187
[5] Coifman R, Meyer Y. Au del`a des op´erateurs pseudo-diff´erentiels. Ast´erisque #57, Soci´et´e Math de France,1978
[6] Deng D G, Yan L X, Yang Q X. Blocking analysis and T(1) theorem. Science in China, 1998, 41: 801–808
[7] Deng D G, Yan L X, Yang Q X. On H¨ormander condition. Chinese Science Bulletin, 1997, 42: 1341–1345
[8] Deng D G, Yan L X, Yang Q X. L2 boundedness of commutators of Calder´on-Zygmund singular integral operators. Progress in Natural Science, 1998, 8(4): 416–427
[9] Fefferman C. Lp bounds for pseudo-differential operators. Israel J Math, 1973, 14: 413–417
[10] Folland G B. Harmonic Analysis on Phase Space. Ann of Math Studies. Princeton, NJ: Princeton University Press, 1989
[11] H¨ormander L. The Analysis of Partial Differential Equation, I, II, III, IV. New York: Springer-Verlag, 1983–1985
[12] Leopold H G, Yang Q X. L2 continuity and general non-smooth H¨ormander symbol operators, preprint.
[13] Meyer Y. Ondelettes et Op´erateurs, I et II. Paris: Hermann, 1991–1992
[14] Meyer Y, Yang Q X. H¨ormander condition and T1 theorems in the general context//Liu Peide, ed. Functional Space Theory and Its Applications. Proceeding of International Conference & 13th Academic Symposium in China. UK: Research Information Ltd UK, 2003: 169–172
[15] Peng L Z, Yang Q X. Predual spaces for Q spaces. Acta Math Sci, 2009, 29B(2): 243–250
[16] Stein E M. Harmonic Analysis–Real Variable Methods, Orthogonality, and Integrals. Princeton: Princeton University Press, 1993
[17] Yang Q X. Fast algorithms for Calder´on-Zygmund singular integral operators. Appl Comp Harmonic Anal, 1996, 3: 120–126
[18] Yang Q X. New wavelet bases and isometric between symbolic operators spaces OpSm 1,δ and kernel distribution spaces, Acta Math Sin, English Series, 2002, 18(1): 107–118
[19] Yang Q X. Wavelet and Distribution. Beijing: Beijing Science and Technology Press, 2002
[20] Yang Q X. Wavelet characterization of H¨ormander symbol class Smρ,δ and applications to the operator’s continuity. Proc Indian Acad Sci (Math Sci), 2005, 115(3): 1–22
[21] Yang Q X. Symbol operator and the speed of approximation by compact operator. Acta Math Sin, Chinese Series, 2007, 50(5)
[22] Yang Q X, Chen Z X, Peng L Z. Uniform characterization of function spaces by wavelets. Acta Math Sci, 2005, 25B(1): 130–144
[23] Yang Z Y, Yang Q X. A note on Convolution-type Calderon-Zygmund operators. Acta Math Sci, 2009, 29B(5): 1341–1350
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