Weighted Lebesgue space with exponential weights: smoothing effect and properties

Question

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ where the weight is exponential, for example of the type $\omega(x) = \exp \left( -c|x|^2 \right)$. 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: $\|e^{-t \Delta} f\|_p \leq c t^{-\frac{n}{2} \left( \frac{1}{q}-\frac{1}{p}\right)} \|f\|_q$, $1\leq q\leq p\leq \infty.$ If anyone knows, can you recommend works and books that deal with the subject? Thanks in advance.

Answer 1

This kind of weight reminds me a lot of Gaussian weights. Then, you can normalize it and put it into the measure to get an $L^p$ space on $\mathbb R^n$ endowed with the Gaussian measure.

My intuition says expect the kind of PDE estimates that you would find in a bounded domain.

And also, the question is related to Malliavin Calculus. Probably people doing SPDEs know more about this than functional analysis and PDE people.

I give you two PDFs that I found that might be useful:

https://www.math.utah.edu/~davar/math7880/S15/Chapter2.pdf

https://diposit.ub.edu/dspace/bitstream/2445/158898/1/158898.pdf