Measure, Integration & Real Analysis · Problem 11C.13

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · poisson-kernel-norm-monotone

Problem

Suppose p∈[1,∞]p \in [1, \infty] and f∈Lp(R)f \in L^p(\mathbf{R}). Prove that if 0<y<y′0 < y < y', then ∥Pyf∥p≥∥Py′f∥p.\|\mathcal{P}_y f\|_p \geq \|\mathcal{P}_{y'} f\|_p.

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