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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2013 • Question 5

Question

Math 6210 • Real Analysis Fall 2013 • Question 5

Let \((X,M,\mu )\) be a \(\sigma \)-finite measure space, and consider \(X\times X\) with the product measure. Suppose that \(K\in L^2(X\times X)\). Show that if \(f\in L^2(X)\), then \(Tf\in L^2(X)\) and \(\lVert Tf\rVert _2\le \lVert K\rVert _2\lVert f\rVert _2\) where \[ Tf(x)=\int _X K(x,y)f(y)\,d\mu (y). \]

Raw LaTeX

Let \((X,M,\mu)\) be a \(\sigma\)-finite measure space, and consider
\(X\times X\) with the product measure. Suppose that \(K\in L^2(X\times X)\).
Show that if \(f\in L^2(X)\), then \(Tf\in L^2(X)\) and
\(\lVert Tf\rVert_2\le \lVert K\rVert_2\lVert f\rVert_2\) where
\[
Tf(x)=\int_X K(x,y)f(y)\,d\mu(y).
\]

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Product Measures and Fubini-Tonelli Theorems