| Summary: | In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established.
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