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Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) such that for all $q>p$ and for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ for all $q>p$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!

Assume that $1<n\leq p$. Does there exist a (positive) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) such that for all $q>p$ and for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ for all $q>p$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) such that for all $q>p$ and for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ for all $q>p$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!

Assume that $1<n\leq p$. Does there exist a (positive) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) such that for $q>p$ and for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ for $q>p$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!

Assume that $1<n\leq p$. Does there exist a (positive) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) such that for all $q>p$ and for all $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, there exists a constant $C=C(n, p)>0$ such that \begin{equation}\left(\int_{\mathbb{R}^{n}}{|f|^{q}d\mu}\right)^{p/q}\leq C\int_{\mathbb{R}^{n}}{|\nabla f|^{p}dx}?\end{equation}

In the book of Maz'ya I found that the above inequality holds in the case $p=1$ provided that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{n-1}}<\infty,$$ and in the case $1\leq p<n$ for all $q>p$ under the condition that $$\sup_{x\in \mathbb{R}^{n}, r>0}\frac{\left(\mu(B(x, r)\right)^{1/q}}{r^{\frac{n}{p}-1}}<\infty.$$

Is there something known in the case $1<n\leq p$?

Thanks in advance!