Abstract
We study expansions of pseudodifferential operators from the Hörmander class in a special family of functions called brushlets. We prove that such operators have a sparse representation in a brushlet system. Using this sparsity, we show that a pseudodifferential operator extends to a bounded operator betweenα-modulation spaces. These spaces were introduced by Gröbner in [15]. They are, in some sense, intermediate spaces between the classical Besov and Modulation spaces.
Highlights
ÂÇÍÊÆ Ä Ç ÍÆ ÌÁÇÆ ËÈ Ë Æ ÈÈÄÁ ÌÁÇÆË3⁄41⁄41⁄41⁄4 Å Ø Ñ Ø × ËÙ Ø Ð ×× ¬ Ø ÓÒo ÈÖ Ñ ÖÝ 1⁄2 Ë ÓÒ ÖÝ 3⁄4 1⁄21⁄4 ̧ ¿1⁄4 ̧ ¿1⁄4 à ÝÛÓÖ × Ò Ô Ö × ×o È× Ù Ó « Ö ÒØ Ð ÓÔ Ö ØÓÖ× ̧ ÑÓ ÙÐ Ø ÓÒ ×Ô × ̧ ÖÙ× Ð Ø×o.
Ì ÔÖÓ Ð Ñ Ó ÒÚ ×Ø Ø Ò ÓÙÒ Ò ×× Ó Ô× Ù Ó « Ö ÒØ Ð ÓÔ Ö ØÓÖ× ́© Ó3×μ Ò Ú Ö ÓÙ× ÙÒ Ø ÓÒ ×Ô × × Ò ÓÒ× Ö Ý ÒÙÑ ÖÓÙ× ÙØ ÓÖ×o ÓÙÖ Ù Ò ÓÒ× 1⁄23⁄4 ÓÒ× Ö Ø ÓÙÒ Ò ×× ÓÒ ×ÓÚ ×Ô × ̧ Ò È Ú Ö ÒØ 3⁄41⁄2 Ò Ù 3⁄43⁄4 ÓÒ ÌÖ Ð1Ä ÞÓÖ Ò ×Ô ×o ÁÒ 3⁄4 ̧ 3⁄4 ̧.
ÑÞÒÖÐÞØ × Ö ×ÙÐØ× Ý ÓÒ× Ö Ò Ô Ö « Ö ÒØ Ð ÓÔ Ö ØÓÖ× Ò Ò ×ÓØÖÓÔ ×ÓÚ Ò ÌÖ Ð1Ä ÞÓÖ Ò ×Ô ×o ÅÓÖ Ö ÒØÐÝ Ø Ö
Summary
3⁄41⁄41⁄41⁄4 Å Ø Ñ Ø × ËÙ Ø Ð ×× ¬ Ø ÓÒo ÈÖ Ñ ÖÝ 1⁄2 Ë ÓÒ ÖÝ 3⁄4 1⁄21⁄4 ̧ ¿1⁄4 ̧ ¿1⁄4 à ÝÛÓÖ × Ò Ô Ö × ×o È× Ù Ó « Ö ÒØ Ð ÓÔ Ö ØÓÖ× ̧ ÑÓ ÙÐ Ø ÓÒ ×Ô × ̧ ÖÙ× Ð Ø×o. Ì ÔÖÓ Ð Ñ Ó ÒÚ ×Ø Ø Ò ÓÙÒ Ò ×× Ó Ô× Ù Ó « Ö ÒØ Ð ÓÔ Ö ØÓÖ× ́© Ó3×μ Ò Ú Ö ÓÙ× ÙÒ Ø ÓÒ ×Ô × × Ò ÓÒ× Ö Ý ÒÙÑ ÖÓÙ× ÙØ ÓÖ×o ÓÙÖ Ù Ò ÓÒ× 1⁄23⁄4 ÓÒ× Ö Ø ÓÙÒ Ò ×× ÓÒ ×ÓÚ ×Ô × ̧ Ò È Ú Ö ÒØ 3⁄41⁄2 Ò Ù 3⁄43⁄4 ÓÒ ÌÖ Ð1Ä ÞÓÖ Ò ×Ô ×o ÁÒ 3⁄4 ̧ 3⁄4 ̧. ÑÞÒÖÐÞØ × Ö ×ÙÐØ× Ý ÓÒ× Ö Ò Ô Ö « Ö ÒØ Ð ÓÔ Ö ØÓÖ× Ò Ò ×ÓØÖÓÔ ×ÓÚ Ò ÌÖ Ð1Ä ÞÓÖ Ò ×Ô ×o ÅÓÖ Ö ÒØÐÝ Ø Ö
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