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Jul
16
asked Is there a specific and useful strategy for this kind of general setup of a problem?
Jul
15
accepted How does this conjecture correspond to Carleson's theorem for the case of $d=1$?
Jul
14
asked How does this conjecture correspond to Carleson's theorem for the case of $d=1$?
Jul
13
comment Is this operator a Fourier multiplier operator?
In my case, $f$ is a generic function, it may contain jump discontinuties. Basically it is a function of bounded variation. So how can I handle this definition in my problem?
Jul
10
comment Is this operator a Fourier multiplier operator?
great! looking forward
Jul
10
comment Is this operator a Fourier multiplier operator?
If you don't find it interesting, then no problem.
Jul
10
comment Is this operator a Fourier multiplier operator?
You know my motivational problem, so why not substitute this expression there and take it forward to see what can be done in proving it, I offer you a joint publication!
Jul
10
comment Is this operator a Fourier multiplier operator?
ok, no problem. I should really thank you for the nice answer.
Jul
10
comment Is this operator a Fourier multiplier operator?
Ah, then I feel $x$ should replace $y$ in the expression sitting inside the Hilbert operator, that would avoid confusion and there really isnt any need for $y$ when we can do with $x$. I mean we cannot write for example $f(x) \ast g(y) \ast h(t)$ right we just have to write $f(t)\ast y(t) \ast h(t)$ . Correct me if I got it wrong. I agree you needed $y$ inside the integral to explain principal value, but later it can be avoided.
Jul
10
comment Is this operator a Fourier multiplier operator?
I don't get what two variables $x$,$y$ doing in this single expression $$\pi e^{i\alpha x} \mathcal{H}(e^{-iy\alpha}f) $$, and $t$ not appearing, may be I assume $x$ was taken instead of $t$ but what is $y$ doing in it. This I think I probably not getting the context here, as I am not having working knowledge on distributions. It would help if you could throw some light on it.
Jul
10
revised Is this operator a Fourier multiplier operator?
added 273 characters in body
Jul
8
comment Is this operator a Fourier multiplier operator?
Sounds good, thanks. Will get back soon as I am on vacation.
Jul
7
revised Is this operator a Fourier multiplier operator?
added 77 characters in body
Jul
7
comment Is this operator a Fourier multiplier operator?
@CameronWilliams Consider $f(t)*\frac{e^{i\alpha t}}{t}$, I want to take a derivative of this and then look at it as a fourier multiplier acting on f, but looks like such a thing is possible only through tempered distribution. Please look at this answer and suggest me something how to move forward.
Jul
7
comment Is this operator a Fourier multiplier operator?
@JoãoRamos Consider $f(t)*\frac{e^{i \alpha t}}{t}$, I want to take a derivative of this and then look at it as a fourier multiplier acting on $f$, but looks like such a thing is possible only through tempered distribution. Please look at this answer and suggest me something how to move forward.
Jul
7
comment Is this operator a Fourier multiplier operator?
@JoãoRamos : I understand the integral won't converge, I want to understand how I can use this distribution conceptually, in my problem. It appears that the rules are different here with distributions than functions (as per my experience) but I dont know the rules.
Jul
7
comment Is this operator a Fourier multiplier operator?
@CameronWilliams : It would be great if you can explain it in an answer, and how to make sense of a distribution.
Jul
7
comment Is this operator a Fourier multiplier operator?
@CameronWilliams : I dont want to compute Fourier transform of $f$, $f$ is a generic function in my analysis. So What you said is true, I want Fourier transform of $L_{\alpha}$ and then I multiply it with FT of $f$. I don't know much about distributions, so I don't get what is not clearer, it appears to me that you have answered my question, but I don't know how to deal with distributions, perhaps that where I am missing the point.
Jul
7
comment Is this operator a Fourier multiplier operator?
@JoãoRamos : Yes, please feel free to use distributions. Ultimately I am tryinng to solve this problem, and I hope distributions can take me there. mathoverflow.net/a/210787/14414
Jul
7
asked Is this operator a Fourier multiplier operator?