Please use this identifier to cite or link to this item: http://prr.hec.gov.pk/jspui/handle/123456789/605
Title: O N THE B OUNDEDNESS AND M EASURE OF N ON - COMPACTNESS FOR M AXIMAL AND P OTENTIAL O PERATORS
Authors: ASIF, MUHAMMAD
Keywords: Natural Sciences
Mathematics
Issue Date: 2008
Publisher: GC University Lahore, Pakistan
Abstract: The essential norm of maximal and potential operators defined on homogeneous groups is estimated in terms of weights. The same problem is discussed for par- tial sums of Fourier series, Poisson integrals and Sobolev embeddings. In some cases we conclude that there is no a weight pair (v, w) for which the given operator is compact from L pw to L qv . It is proved that the measure of non-compactness of a bounded linear operator from a Banach space into a weighted Lebesgue space with variable parameter is equal to the distance between this operator and the class of finite rank operators. The p(x) essential norm of the Hilbert transform acting from L w p(x) to L v is estimated from below. As a corollary we have that there is no a weight pair (v, w) and a function p from the class of log-H ╠łolder continuity such that the Hilbert transform is compact p(x) from L w p(x) to L v . Necessary and sufficient conditions on a weight pair (v, w) governing the bound- edness of generalized fractional maximal functions and potentials on the half-space q(x) from L pw (R n ) to L v (R n+1 + ) are derived. As a corollary, we have criteria for the trace inequality for these operators in variable exponent Lebesgue spaces.
URI:  http://prr.hec.gov.pk/jspui/handle/123456789//605
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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