Applications of Multipliers to the Theory of Integral Operators
In this chapter it is shown that Sobolev multipliers are useful for the study of integral operators. First, in Sect. 16.1 we consider an arbitrary convolution operator acting in a pair of weighted L2-spaces and collect corollaries of the theory of multipliers providing criteria of boundedness and compactness of the convolutions and a characterization of their spectra. Next we turn to classical singular integral operators acting in Sobolev spaces. In Sect. 16.2 a calculus of these operators is developed under the assumption that their symbols belong to classes of multipliers in Sobolev spaces. Finally, in Sect. 16.3 sharp conditions for continuity of the singular integral operators acting from W m 2 to W l 2 are found. These conditions are formulated in terms of certain classes of multipliers.
KeywordsSobolev Space Integral Operator Singular Integral Operator Convolution Operator Singular Operator
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