I am researching whether there are weighted Lebesgue spaces of the type $$ \{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_{\omega}}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\} $$$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ where the weight is exponential, for example of the type $\omega(x) = \exp{ \left( -c|x|^2 \right)}$$\omega(x) = \exp \left( -c|x|^2 \right)$. So far I haven't found anything, whether such spaces exist, and what their properties are. I'm looking for these spaces to try to verify the existence of PDE's (in these weighted spaces) of the type $u_t - \Delta u = f(x)$ where $f(x) \rightarrow \infty$ when $|x| \rightarrow \infty.$ In particular, it would be interesting to obtain regularizing estimates of the semigroup such as those of the semigroup in the usual $L^p$ spaces: $||e^{-t \Delta}f||_p \leq c t^{-\frac{n}{2} \left( \frac{1}{q}-\frac{1}{p}\right)}||f||_q$$\|e^{-t \Delta} f\|_p \leq c t^{-\frac{n}{2} \left( \frac{1}{q}-\frac{1}{p}\right)} \|f\|_q$, $1\leq q\leq p\leq \infty.$ If anyone knows, can you recommend works and books that deal with the subject? Thanks in advance.
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