| Summary: | In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>M</mi><mrow><mi>p</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>p</mi></msub><mo>,</mo><mi>p</mi><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> As an application, we prove the compactness of the commutator of the Riesz potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub><mo>]</mo></mrow></semantics></math></inline-formula> in generalized Morrey spaces, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>V</mi><mi>M</mi><mi>O</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mi>M</mi><mi>O</mi><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> denote the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>M</mi><mi>O</mi></mrow></semantics></math></inline-formula>-closure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>C</mi><mn>0</mn><mo>∞</mo></msubsup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>). We prove auxiliary statements regarding the connection between the norm of average functions and the norm of the difference of functions in the generalized Morrey spaces. Such results are also of independent interest.
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