Lp CONTINUITY OF HÖRMANDER SYMBOL OPERATORS OpSm0,0 AND NUMERICAL ALGORITHM

doi:10.1016/S0252-9602(11)60338-X

Abstract

Abstract:

If we use Littlewood-Paley decomposition, there is no pseudo-orthogonality for H¨ormander symbol operators O_pS^m_0,0, which is different to the case S^m_ρ,δ (0 ≤δ < ρ ≤ 1). In this paper, we use a special numerical algorithm based on wavelets to study the L^pcontinuity of non infinite smooth operators O_pS^m_0,0; 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→L²continuity for scale operators. By considering the influence region of scale operator, we get H¹(= F^0,2₁ ) → L¹ continuity and L¹ →BMO continuity. By interpolation theorem, we get also L^p(= F^0,2_p ) → L^p continuity for 1 < p < ∞. Our results are sharp for F^0,2_p →L^pcontinuity 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.

Key words: H¨ormander symbol class, wavelet and numerical algorithm, basic operators and scale operators, approximation by compact operator and operator’s continuity

CLC Number:

42B20

YANG Qi-Xiang. L^p CONTINUITY OF HÖRMANDER SYMBOL OPERATORS O_pS^m_0,0 AND NUMERICAL ALGORITHM[J].Acta mathematica scientia,Series B, 2011, 31(4): 1517-1534.

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L^p CONTINUITY OF HÖRMANDER SYMBOL OPERATORS O_pS^m_0,0 AND NUMERICAL ALGORITHM

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