Estimating the integral in norm.

I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively. But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$.

I tried using Hölder but I am not given any more information. Even if I could apply Hölder then I always get confused on introducing $r$. I know its quite easy but I think a small hint could do if I have to apply Hölder.

Thanks!

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Is there a link between $p,q$ and $r$ or not? – Davide Giraudo Nov 22 '12 at 19:22
@Davide Giraudo : This is a question thrown out by my classmate. I think $p^{-1}+q^{-1}=1$ should be the criteria right ?? If thats the case can you give me a small hint to apply Hölder inequality ?? – Theorem Nov 22 '12 at 19:26
@Davide Giraudo : if $r=1$ then i would apply it straight forward, but if r is not $1$ i get often confused in choosing the right exponents . – Theorem Nov 22 '12 at 19:27
Maybe the relationship is $\frac 1r=\frac 1p+\frac 1q$. – Davide Giraudo Nov 22 '12 at 20:24
@DavideGiraudo : Sir, I am pretty sure i was just told that $r\in[1,\infty]$, and had no relation with $p$ and $q$. – Theorem Nov 22 '12 at 20:28

1 Answer

Note that if $f \in L^q(A)$ and $g \in L^q(B)$, $f(x) g(y) \in L^q(A \times B)$. Then if $k \in L^p(A \times A)$ with $1/p + 1/q = 1$, $$\left| \int_{A \times B} k(x,y) f(x) g(y)\ dx \ dy \right| \le \|k\|_p \|f\|_q \|g\|_q$$ Since this is true for every $g \in L^q(B)$, $\int k(x,y) f(x)\ dx \in L^p(B)$ with $\|\int k(x,y) f(x)\ dx \|_p \le \|k\|_p \|f\|_q$.

However, it won't work if $r \ne p$. In order to have $\int_A k(x,y) f(x)\ dx \in L^r(B)$, what you'd want is $k(\cdot,y) \in L^p$ for almost all $y \in B$ with $y \to \|k(\cdot,y)\|_p \in L^r(B)$. If $L^p(B)$ is not a subset of $L^r(B)$, this won't be true. In fact, given $h \in L^p(B) \backslash L^r(B)$ and $f\ne 0 \in L^q(A)$, take $g \in L^p(A)$ with $\|g\|_p = 1$ and $\int_A f g \ dx = \|f\|_q$, and define $k(x,y) = g(x) h(y)$. Then $k \in L^p$ with $\|k\|_p = \|g\|_p \|h\|_p$, and $\int_A k(x,y) f(x)\ dx = \|f\|_q h(y) \notin L^r(B)$.

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