Timeline for Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Current License: CC BY-SA 4.0
12 events
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| May 5, 2020 at 17:22 | comment | added | JohnA | The fault is mine. My original question did not mention the requirement that $\int|f-g|^p\to0$, and while your counterexample is nice, it misses the main intent of my question. As I suggest above, we can merge @WillieWong's comment with your answer and this would be satisfactory. | |
| May 5, 2020 at 0:00 | comment | added | Iosif Pinelis | Previous comment continued: Yet another, rather different kind of studies where we have $\|f\|_p=O(\|f\|_q)$ with $q\in(0,p)$ is represented by so-called reverse Hölder's inequalities; see e.g. inequality (3.6) in ams.org/journals/tran/1993-340-01/S0002-9947-1993-1124164-0/… | |
| May 4, 2020 at 23:51 | comment | added | Iosif Pinelis | Previous comment continued: In probability, cases of the inequality $\|f\|_p=O(\|f\|_q)$ with $q\in(0,p)$ are referred to as the hypercontractivity phenomenon -- see e.g. projecteuclid.org/euclid.aop/1176990550 and goodreads.com/book/show/13708363-introduction-to-random-chaos | |
| May 4, 2020 at 23:50 | comment | added | Iosif Pinelis | @JohnA : Your question was fully answered. Your new question, concerning Ladyzhenskaya's inequalities (which also involve norms of the gradient), is a wholly different thing. If you now want to ask a general question as to when we have (say) $\|f\|_p=O(\|f\|_q)$ with $q\in(0,p)$, then this question is way too broad to be asked on MathOverflow. | |
| May 4, 2020 at 21:15 | history | edited | LSpice | CC BY-SA 4.0 |
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| May 4, 2020 at 20:50 | comment | added | Willie Wong | @JohnA: Ladyzhenskaya assumes a lot more. In effect it is assuming that your sequence of function converges in $L^p$ and $L^r$, and so necessarily converges in $L^q$ when $q\in (p,r)$. Additionally, in this setting if you have rates in $L^p$, then by interpolation you always get some rate in $L^q$, even without knowing anything about the rate of convergence in $L^r$. | |
| May 4, 2020 at 18:31 | comment | added | JohnA | It's probably better think of my question as asking "under what reasonable assumptions is such a bound possible". I edited my question to point out an example in Ladyzhenskaya's inequality, although this assumes a bit more than I wanted. (I am not sure about your point on $\alpha_n$, this is simply the rate of convergence.) | |
| May 3, 2020 at 18:08 | comment | added | Iosif Pinelis | @JohnA : I am still not sure about what specifically you mean by "arbitrarily bad". Can you just state it in formal terms? Also, I see no point in saying $c\alpha_n\to0$, because you can always rescale $f_n$ and $f$ (by replacing them, say, by $f_n/(c\alpha_n)^{1/q}$ and $f/(c\alpha_n)^{1/q}$) to get $c\alpha_n$ replaced by $1$. Alternatively, you can similarly rescale $f_n$ and $f$ to get $\int|f-g|^p=\int|(f^p-g^p)=1$, as was done in my example. | |
| May 3, 2020 at 17:33 | comment | added | JohnA | My comment was vague so my apologies; please see my edits to the "example" in my post. I'm curious if there is any relationship between the rate of convergence in $L^q$ vs $L^p$. Again, I suspect, the answer is no, but I am not sure. | |
| May 3, 2020 at 16:56 | comment | added | Iosif Pinelis | @JohnA : I am not sure how to understand your comment. Can you specify what kind of bound you want to prove or disprove? | |
| May 3, 2020 at 15:35 | comment | added | JohnA | This is a great counterexample. Do you have any idea if this extends to the case where either (a) $\int f^p-\int g^p \to 0$ or (b) $\int |f-g|^p \to 0$? | |
| May 3, 2020 at 15:22 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |