$ ( \int_{\mathbb R}[\int_{\mathbb R} |f(x,y)|^{p} dx]^{p/q} dy)^{1/q} < \infty$ for large $N$?

Question

Suppose that $f:\mathbb R^{2} \to \mathbb R$ such that $$f(x,y)=\frac{1}{(1+ r)^N},$$ where $r=\sqrt{x^{2}+y^{2}}.$

Let $1\leq p, q < \infty.$

My Question is: Can we expect: $ ( \int_{\mathbb R}[\int_{\mathbb R} |f(x,y)|^{p} dx]^{q/p} dy)^{1/q} < \infty$ for large $N$?

Answer 1

For $x,y\in\mathbb{R} $ the followin inequality holds true $$(1+\sqrt{x^2 +y^2 ) }\geqslant \sqrt[4]{(1+x^2 )(1+y^2)}$$ hence $$( \int_{\mathbb R}[\int_{\mathbb R} |f(x,y)|^{p} dx]^{q/p} dy)^{1/q} \leqslant( \int_{\mathbb R}[\int_{\mathbb R} [(1+x^2)(1+y^2)]^{-\frac{Np}{4}} dx]^{q/p} dy)^{1/q} =\left(\int_{\mathbb{R}}(1+y^2)^{-\frac{Nq}{4}} dy\right)^{\frac{1}{q}}\cdot \left(\int_{\mathbb{R}}(1+x^2)^{-\frac{Np}{4}} dx\right)^{\frac{1}{p}} <\infty$$ for large $N.$