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Post Closed as "Not suitable for this site" by Gro-Tsen, Michael Renardy, Mark Wildon, Alexandre Eremenko, Amir Sagiv
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dohmatob
dohmatob
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Is SchwarzSchwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the SchwarzSchwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=1$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^1(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking $m$ sufficiently larger than $n$, we see that $f \in H^1(\mathbb R^n)$.

Is Schwarz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwarz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=1$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^1(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking $m$ sufficiently larger than $n$, we see that $f \in H^1(\mathbb R^n)$.

Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=1$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^1(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking $m$ sufficiently larger than $n$, we see that $f \in H^1(\mathbb R^n)$.

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dohmatob
dohmatob
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Can someone kindly confirm that the Schwarz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=2$$k=1$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^2(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$$$ \|f\|_{H^1(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking sufficiently $m$ sufficiently larger than $n$, we see that $f \in H^2(\mathbb R^n)$$f \in H^1(\mathbb R^n)$.

Can someone kindly confirm that the Schwarz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=2$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^2(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking sufficiently $m$ sufficiently larger than $n$, we see that $f \in H^2(\mathbb R^n)$.

Can someone kindly confirm that the Schwarz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=1$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^1(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking $m$ sufficiently larger than $n$, we see that $f \in H^1(\mathbb R^n)$.

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dohmatob
dohmatob
  • 6.9k
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  • 18
  • 76

Is Schwarz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwarz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.

After all, at least for the case $k=2$, it holds for any $f \in \mathcal S(\mathbb R^n)$ and $m>0$ that

$$ \|f\|_{H^2(\mathbb R^n)}^2 = \int_{\mathbb R^n}Lap(f)(x)\,dx \le C\int_{\mathbb R^n}(1+\|x\|)^{-m}\,dx \le C'\int_0^\infty (1+r)^{-m}r^{n-1}\,dr. $$ Thus, taking sufficiently $m$ sufficiently larger than $n$, we see that $f \in H^2(\mathbb R^n)$.