The Approximation Functional and Interpolation with Perturbed Continuity |
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Authors: | Y Sagher P Shvartsman |
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Institution: | Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois, 60607, f1;Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel, f2 |
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Abstract: | The continuity conditions at the endpoints of interpolation theorems, TaBjMj aAj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B1, B0)L∞M0 aA0 and E(t, Ta; B0, B1)L∞M1 aA1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L∞ norms can be replaced by BMO(
+) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL0, i.e., f*L∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO(
+). |
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Keywords: | interpolation of operators weak type classes change of measure approximation functional |
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